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Full text of "A commentary on Newton's Principia.: with a supplementary volume"

JULIUS WANODflfflEIM 87 




Mathematics Dept 



COMMENTARY 



NEWTON S PRINCIPIA. 



A SUPPLEMENTARY VOLUME. 



DESIGNED FOR THE USE OF STUDENTS AT THE UNIVERSITIES. 



J. M. F. WRIGHT, A. B. 

LATE SCHOLAR OF TRINITY COLLKGK, CAMBRIDGE, AUTHOR OF SOLUTIO.NS 
OF THE CAMBRIDGE PROBLEMS, &c. &C. 



IN TWO VOLUMES. 
VOL. II. 



LONDON: 
PRINTED FOR T. T. & J. TEGG, 73, CHEAPSIDE; 

AND RICHARD GRIFFIN & CO., GLASGOW. 



MDCCCXXXIII. 






.. 

: 





v : . * 



c^ 




9- 



GLASGOW: 

GEORGE BROOKMAX, PHINTER, VILLAFIELU. 



INTRODUCTION 



VOLUME II. 



AND TO THE 



MECANIQUE CELESTE. 



ANALYTICAL GEOMETRY 

1. To determine the position of a point in Jixed space. 

Assume any point A in fixed space as known and immoveable, and let 




Z z 



three fixed planes of indefinite extent, be taken at right angles to one 
another and passing through A. Then shall their intersections A X , 
A Y , A Z pass through A and be at right angles to one another. 



ii INTRODUCTION. 

This being premised, let P be any point in fixed space ; from P draw 
P z parallel to A Z, and from z where it meets the plane- X A Y, draw 
z x, z y parallel to A Y, AX respectively. Make 
Ax = x, A y = y, P z = z. 

Then it is evident that if x, y, z are given, the point P can be found 
practically by taking A x = x, A y = y, drawing x z, y z parallel to 
AY, AX; lastly, from their intersection, making z P parallel to A Z 
and equal to z. Hence x, y, z determine the position of the point P. 

The lines x, y, z are called the rectangular coordinates of the point P ; 
the point A the origin of coordinates ; the lines A X, A Y, A Z the axes 
of coordinates, A X being further designated the axis of x, AY the axis 
of y, and A Z the axis of z ; and the planes X A Y, Z A X, Z A Y co 
ordinate planes. 

These coordinate planes are respectively denoted by 
plane (x, y), plane (x, z), plane (y, z) ; 
and in like manner, any point whose coordinates are x, y, z is denoted 

briefly by 

point (x, y, z). 

If the coordinates x, y, z when measured along AX, AY, A Z be 
always considered positive; when measured in the opposite directions, 
viz. along A X A Y , A Z , they must be taken negatively. Thus ac 
cordingly as P is in the spaces 

ZAXY, ZAYX , ZAX Y , ZAY X; 
Z ; AXY, Z AYX , Z AX Y , Z AY X, 
the point P will be denoted by 

point (x, y, z), point ( x, y, z), point ( x, y, z), point (x, y, z) 5 
point (x, y, - z), point (- x, y, - z), point (- x, - y, - z), point (x, - y, - z) 
respectively. 

2. Given the position of two points (a, ft 7), ( , , /) in Jixed space, 
to find the distance let-ween them. 

The distance P P is evidently the diagonal of a rectangular parallele 
piped whose three edges are parallel to A X, A Y, A Z and equal to 

a s a , (S s j8 , 7 s /. 
Hence 

P P = V (a a ) 2 + (ft (3 ) 2 + (7 /)* 0) 

the distance required. 

Hence if P coincides with A or a , /S , 7 equal zero, 

P A = V~^ z + /3 2 + 7* (2) 



ANALYTICAL GEOMETRY. iii 

3. Calling the distance of any point P (x, y, z) from the origin A of 
coordinates the radius-vector, and denoting it by g, suppose it inclined to 
the axes AX, AY, A Z or to the planes (y, x}, (x, z), (x, y), by the 
angles X, Y, Z. 

Then it is easily seen that 

x = g cos. X, y = g cos. Y, z = g cos. Z (3) 

Hence (see 2) 

cos. X = ,. 2~r r~i \ > cos - Y = / / z I 2 _i_ i\ 



so that when the coordinates of a point arc given, the angles which the ra~ 
dins-vector makes with each of the axes may hence be found. 

Again, adding together the squares of equations (3), we have 
( x 4. y 2 + z 2 ) = g 2 (cos. 2 X + cos. 2 Y + cos. 2 Z). 
But 

g 2 = x 2 + y 8 + z 2 (see 2), 

.-. cos. 2 X + cos. 2 Y + cos. 2 Z = 1 . . . . . (5) 
which shows that when two of these angles are given the other may be 
found. 

4. Given two points in space, viz. (a, {3, 7), (of, (B f , 7 ), and one of the 
coordinates of any other point (x, y, z) in the straight line that passes 
through them, to determine this other point , that is, required the equations 
to a straight line given in space. 

The distances of the point (a, /3, 7) from the points (a , /3 , 7 ), and 
(x, y, z) are respectively, (see 2) 

P P = V (a )* + (0 /3 7 ) 1 -f- (7 /)% 
and 

PQ= V (a x) 2 + (0 y) + (7 z) . 
But from similar triangles, we get 

( 7 -z) 2 : (PQ) 2 :: (7 - /) 2 : (P F) 8 
whence 



which gives 

^ a _ )*+ (/ 3_/30 2 H7_z)*=(7-7 )U( x) 2 + (^-y) 2 } 

But since a, a! are independent of /3, $ and vice versa, the two first 
terms of the equation, 



v INTRODUCTION. 

are essentially different from the last. Consequently by (6 vol. 1.) 



which give 



_/3 ) 2 ( 7 -z) 2 = ( 7 -/) 2 0- 



z 7 = + 



(6) 



These results may be otherwise obtained; thus, pgp ,is the projection 
of the given line on the plane (x, y) &c. as in fig- 



P 




Hence 



Also 



P f l P 



:/ y::pq:pp 
: : m n : m p 

: : y - /3 : /3 _ 



z Y: / 7 : : p q : p p : : p r : p m 

: : a X : a a . 

Hence the general forms of the equations to a straight line given in 
space, not considering signs, are 



z = x + b\ 
f 



z = a y + b 
To find where the straight line meets the planes, (x, y), (x, z), (y, z), 

we make 

z = 0, y = 0, x = 0, 

which give 






ANALYTICAL GEOMETRY. 
b 



z = b 

. b/ ~ b 
a 

z = b 

b b 

y : ~V~ 
which determine the points required. 

To find when the straight line is parallel to the planes, (x, y), (x, z), 
(y, z), we must make z, y, x, respectively constant, and the equations be 
come of the form 

2 = C 

a y = ax + b b 

To find when the straight line is perpendicular to the planes, (x, y), 
(x, z) (y, z), or parallel to the axes of z, y, x, we must assume x, y ; 
x, z ; y, z; respectively constant, and z, y, x, will be any whatever. 

To find the equations to a straight line passing through the origin of 
coordinates ; we have, since x = 0, and y = 0, when z = 0, 



(9) 
z = a yj 

5. Tojind the conditions that two straight lines in fixed space may inter- 
Sect one another ; and also their point of intersection. 

Let their equations be 

z = ax + A 
z = by + B 
z = a x + A 1 
z = b y + B f 

from which eliminating x, y, z, we get the equation of condition 
a A a A _ b B b B 
a a b b 

Also when this condition is fulfilled, the point is found from 

z = a A ,-" A . (10) 

a a 

6. Tojind the angle /, at which these lines intersect. 

Take an isosceles triangle, whose equal sides measured along these 
lines equal 1, and let the side opposite the angle required be called i ; 
then it is evident that 

cos. I = 1 | i 2 
as 



vi INTRODUCTION. 

But if at the extremities of the line i, the points in the intersecting lines 
be (x , y , z ), (x", y", z"), then (see 2) 

i 2 = (x x") 2 + (y y ) 2 + (z z") 2 
.-. 2 cos. I = 2 J(x x") 2 + (y y") * + (z z") 2 ] 
But by the equations to the straight lines, we have (5) 

z = a x -f A ") 
z =by + BJ 

z" a x" + A \ 
z"=b y + B J 

and by the construction, and Art. 2, if (x, y, z) be the point of intersec 
tion, 

( X _x )*+ (y_y )* + (z z) 2 = 

(x x") 2 + (y y ) 2 + (z z") 2 = 

Also at the point of intersection, 

z = ax+A = a x + A ) 
z = by + B = b y + B J 

From these several equations we easily get 
z z = a (x a ) 

, a M 

y y =-- (x x ) 

z z" = a (x x") 



whence by substitution, 

/ x x /\ 2 i 2 / x x /\ _i_ (x x ) * = 

( X _ x")* + a 2 ( X x") ! + ^ (x x") 2 = 

which give 

x x = 



/* ft X 

/fl + a /J + ^ 

W v T b /2 

Hence 



ANALYTICAL GEOMETRY. vii 

Also, since 

y y = - (x x ) 



arid 

z z = a (x x ) 
z z" = a (x x") 
we have 

a 2 1 . a /2 1 aa 



Hence by adding these squares together we get 
2 cos. 1=2- 



which gives 

1 + aa +^ 
cos.I= - - 25 - _ ..... (11; 



Ttiis result may be obtained with less trouble by drawing straight lines 
from the origin of coordinates, parallel to the intersecting lines ; and then 
finding the cosine of the angle formed by these new lines. For the new 
angle Is equal to the one sought, and the equations simplify into 

z =ax =by , z" = a x" = b / 

z = a x = b y , z=a x =b y 



i j 

From the above general expression for the angle formed by two inter 
secting lines, many particular consequences may be deduced. 

For instance, required the conditions requisite that two straight lines 
given in space may intersect at right angles. 

That they intersect at all, this equation must be fulfilled, (see 5) 
a A a A 7 b B b B ; 
a a "" b b 



viii INTRODUCTION. 

and that being the case, in order for them to intersect at right angles 

t O fD 

we have 



T It 

1 = , cos. 1 = 

and therefore 

a a 



7. In the preceding No. the angle between two intersecting lines is 
expressed in a function of the rectangular coordinates, which determine 
the positions of those lines. " But since the lines themselves would be 
given in parallel position, if their inclinations to the planes, (x, y), (x, z), 
(y, z), were given, it may be required, from other data, to find the same 
angle. 

Hence denoting generally the complements of the inclinations of a 
straight line to the planes, (x, y), (x, z), (y, z), by Z, Y, X, the problem 
may be stated and resolved, as follows : 

Required the angle made by the two straight lines, whose angles of inclina 
tion are Z, Y, X; Z , Y/, X . 

Let two lines be drawn, from the origin of the coordinates, parallel 
to given lines. These make the same angles with the coordinate planes, 
and with one another, as the given lines. Moreover, making an isosceles 
triangle, whose vertex is the origin, and equal sides equal unity, we have 
as in (6), 

cos. I = 1 A i 2 = 1 i( x x ) 2 + (y y ) 2 + (z z ) 2 ? 

the points in the straight lines equally distant from the origin being 
(x, y, z), (x , y , z ). 

But in this case, 

x 2 4. y 2 -f z 2 = 1 

x/ 2 + y 2 + z/ ~ i 

and 

x cos. X, y = cos. Y, z = cos. Z 
x = cos. X , y = cos. Y , z = cos. Z 
. cos. I = x x + y y + z z 

= cos. X. cos. X + cos. Y. cos. Y + cos. Z. cos. Z . . (13) 
Hence when the lines pass through the origin of coordinates, the same 
expression for their mutual inclination holds good ; but at the same time 
X, Y, Z ; X , Y , Z , not only mean the complements of the inclinations 
to the planes as above described, but also the inclinations of the lines to 
the axes of coordinates of x, y, z, respectively. 



ANALYTICAL GEOMETRY. ix 

8. Given the length (L) of a straight line and the complements of its in 
clinations to the planes (x, y), (x, z), (y z), viz. Z, Y, X, tojind the lengths 
of its projections upon those planes, 

By the figure in (4) it is easily seen that 

L projected on the plane (x, y) = L. sin. Z~\ 

(x, z) = L. sin. Y I . . . (14) 

(y, z) = L . sin. X ) 

9. Instead of determining the parallelism or direction of a straight line 
in space by the angles Z, Y, X, it is more concise to do it by means of 
Z (for instance) and the angle d which its projection on the plane (x, y) 
makes with the axis of x. 

For, drawing a line parallel to the given line from the origin of the co 
ordinates, the projection of this line is parallel to that of the given line, 
and letting fall from any point (x, y, z) of the new line, perpendiculars 
upon the plane (x, y) and upon the axes of x and of y, it easily appears, 
that 

x r= L cos. X = (L sin. Z) cos. 6 (see No. 8) 
y = L. cos. Y = (L. sin. Z) sin. 6 
which give 

cos. X = sin. Z. cos. 6\ 

cos. Y = sin. Z . sin. 0) ( 

Hence the expression (13) assumes this form, 

cos. I = sin. Z . sin. Z (cos. 6 cos. 6 + sin. 6 sin. 6 ) + cos. Z cos. Z 

= sin. Z . sin. Z cos. (6 6 ) + cos. Z . cos. Z . . . . (16) 
which may easily be adapted to logarithmic computation. 
The expression (5) is merely verified by the substitution. 

10. Given the angle of intersection (I) between two lines in space and 
their inclinations to the plane (x, y), tojind the angle at which their pro 
jections upon that plane intersect one another. 

If, as above, Z, Z be the complements of the inclinations of the lines 
upon the plane, and d, (f the inclinations of the projections to the axis of 
x, we have from (16) 

cos. ( _ I) = cos. I- cos. Z cos. Z 

sin. Z . sin. Z v 

This result indicates that I, Z, Z are sides of a spherical triangle 
(radius = 1), (f being the angle subtended by I. The form may at 
once indeed be obtained by taking the origin of coordinates as the center 
of the sphere, and radii to pass through the angles of the spherical tri 
angle, measured along the axis of z, and along lines parallel to the 
given lines. 



x INTRODUCTION. 

Having considered at some length the mode of determining the posi 
tion and properties of points and straight lines in fixed space, we proceed 
to treat, in like manner, of planes. 

It is evident that the position of a plane is fixed or determinate in posi 
tion when three of its points are knowiL Hence is suggested the follow 
ing problem. 

11. Having given the three points (a, ]3, y), (a , Q 9 /), (a", 0", / ) in 
space, tojitid the equation to the plane passing through them ; that is, to 
Jind the relation between the coordinates of any other point in the plane. 

Suppose the plane to make with the planes (z, y), (z, x) the intersec 




tions or traces B D, B C respectively, and let P be any point whatever 
in the plane ; then through P draw P Q in that plane parallel to B D, 
&c. as above. Then 

z QN = PQ = QQ cot. D B Z 

= y cot. D B Z. 
But 

QN = AB NA. cot. C B A 

= A B + x cot. C B Z, 
.-. z = A B + x cot. C B Z + y cot. D B Z. 

Consequently if we put A B = g, and denote by (X, Z), (Y, Z) the 
inclinations to A Z of the traces in the planes of (x, z), (y, z) respectively, 
we have 

z = g + x cot. (X, Z) + y cot. (Y, Z) . , . . (18) 
Hence the form of the equation to the plane is generally 



ANALYTICAL GEOMETRY. . xi 

Now to find these constants there are given the coordinates of three 
points of the plane ; that is 

7 = A +B/3 +C 
/ = A a! + B /3 + C 
7" = A a" + B /3" + C 
from which we get 

A - 7/3 / -//3 + /"--/ / /3 + 7"/3-7ff / _ cot / x z) . 

A - a /3 - /3 + a ^ a"? + "/3_/3" - 

R _ r y a + 7 " y" a + y" y " _ t /Y z^ 

B - ^F_a /3 + /3"-"/3 + "/3 ^ - 

_ /3"(y a /) + g(/a" - / ) + ^ (/ 7 - 7 7/ ) 

a /3/ __ / ^ + a |8" a" /3 + a" /3 /3" 
Hence when the trace coincides with the axis of x, we have 

C = 0, 
and 

A = cot. | = 

/3" (7 a / a) + J3 (/ a." / a ) + F (/ 7 a") = > 
7 /3 - / /3 + / j9" / /3 + 7" /3 _ 7 /3" = j 

1 (/3-/3^) . (/ a"- y" a ) + (B ff") . (y" a 7 a^) 
! - " X " " 



a 



and the equation to the plane becomes 

z = By .......... (25) 

When the plane is parallel to the plane (x, y), 

A = 0, B = 0, 
and 

z = C ............. (26) 

from which, by means of A = 0, B = 0, any two of the quantities 7, 7 , y" 
being eliminated, the value of C will be somewhat simplified. 

Hence also will easily be deduced a number of other particular results 
connected with the theory of the plane, the point, and the straight line, of 
which the following are some. 

To find the projections on the planes (x, y), (x, z), (y, z) of the intersec 
tion of the planes, 

z=Ax + By + C, 
z = A x + B y+ C . 
Eliminating z, we have 

(A A )x + (B B )y + C C = .... (27) 
which is the equation to the projection on (x, y). 



xii INTRODUCTION. 

Eliminating x, we get 

(A A)z + (AB A B)y + AC A C = .... (28) 
which is the equation to the projection on the plane (y, z). 
And in like manner, we obtain 

(B B)z + (A B AB )x+ BC B C = . . . . (29) 
for the projection on the plane (x, z). 

To find the conditions requisite that, a plane and straight line shall be 
parallel or coincide. 

Let the equations to the straight line and plane be 
x = a z + A^ 
y = bz + BJ 
z = A x + B y + C . 
Then by substitution in the latter, we get 

z(A a+ B b 1) + A A+ B B + C = 0. 

Now if the straight line and plane have only one point common, we 
should thus at once have the coordinates to that point. 

Also if the straight line coincide with the plane in the above equation, 
z is indeterminate, and (Art. 6. vol. I,) 

A a + B b 1 = 0, A A + B B + C = . . . (27) 

But finally if the straight line is merely to be parallel to the plane, the 

above conditions ought to be fulfilled even when the plane and line are 

moved parallelly up to the origin or when A, B, C are zero. The only 

condition in this case is 

A a + B b = 1 (28) 

To Jtnd the conditions that a straight line be perpendicular to a plane 
z = Ax+By + C. 

Since the straight line is to be perpendicular to the given plane, the 
plane which projects it upon (x, y) is at right angles both to the plane 
(x, y) and to the given plane. The intersection, therefore, of the plane 
(x, y) and the given plane is perpendicular to the projecting plane. Hence 
the trace of the given plane upon (x, y) is perpendicular to the projec 
tion on (x, y) of the given straight line. But the equations of the traces 
of the plane on (x, z), (y, z), are 

z= Ax + C, z = By + 

or 



z = A x -f- L,, z 15 y -f- ^\ 

1 C 1 C( 

x - A z ~ A y ~B Z ~ B) 



(29) 



and if those of the perpendicular be 

x = a z + A,\ 
y = bz + B,J 



ANALYTICAL GEOMETRY. xiii 

it is easily seen from (11) or at once, that the condition of these traces 
being at right angles to the projections, are 
A + a = 0, A + b = 0. 

To draw a straight line passing through a given point (, /3, 7) at right 
angles to a given plane. 

The equations to the straight line, are clearly 

x _ a + A (z 7) = 0, y + B (z 7 ) = 0. . . . (30) 
To find the distance of a given point (a, /3, y) from a given plane. 
The distance is (30) evidently, when (x, y, z) is the common point in 
the plane and perpendicular 



But the equation to the plane then also subsists, viz. 

from which, and the equations to the perpendicular, we have 

z 7= C 7 + A a + B/?, 
therefore the distance required is 

C 7 + A + B 



(31) 



A 2 + B 2 

To find the angle I formed by two planes 
z = Ax + By+C, 
z = A x + B y + C . 

If from the origin perpendiculars be let fall upon the planes, the angle 
which they make is equal to that of the planes themselves. Hence, if 
generally, the equations to a line passing through the origin be 
x =r a z ) 
y = bz/ 

the conditions that it shall be perpendicular to the first plane are 
A + a = 0, 
B + b = 0, 
and for the other plane 

A + a = 0, 
B + b = 0. 
Hence the equations to these perpendiculars are 

x + A z = 
y + Bz = 
x + A z = 

y 



z = \ 

y z = o, J 



xiv INTRODUCTION. 

which may also be deduced from the forms (30). 
Hence from (11) we get 

T _ _ 1 + A A + B B 

J = " (32 > 



Hence to find the inclination (s) of a plane with the plane (x, y). 
We make the second plane coincident with (x, y), which gives 

A = 0, B = 0, 
and therefore 

COS - i= V(1 + A- + B-) ...... (S3) 

In like manner may the inclinations (), (?j) of a plane 
z = Ax + By + C 

to the planes (x, z), (y, z) be expressed by 

COS -^V(l+A* + B*)j ...... (34) 

cos< " = V(l + A 2 + BV 
Hence 

cos. 2 s + cos. 2 + cos. 2 j = 1 ...... (35) 

Hence also, if E , , 53 be the inclinations of another plane to (x, y)> 

(x, z), (y, z). 

COS. I = COS. COS. s + COS. COS. < + COS. 1) COS. Jj . . . (36) 

Tojind the inclination vofa straight line x = a z + A , y = b z + B , 
ft? the plane z = Ax + By+C. 

The angle required is that which it makes with its projection upon the 
plane. If we let fall from any part of the straight line a perpendicular 
upon the plane, the angle of these two lines will be the complement of v. 
From the origin, draw any straight line whatever, viz. x = a z, y = b z. 
Then in order that it may be perpendicular to the plane, we must have 

a = A, b = B. 

The angle which this makes with the given line can be found from (11) ; 
consequently by that expression 

1 A a B b , q7 N 

sm -"= V(i +a* + b*) v(l + A+ B J 

Hence we easily find that the angles made by this line and the coor 
dinate planes (x, y), (x, z), (y, z), viz. Z, Y, X are found from 

^ 1 

cos L - J 



ANALYTICAL GEOMETRY. xv 

cos. Y = , i t i \ g\ > 

cos. X = ^jj a &2 bt> (38) 

which agrees with what is done in (3). 

TRANSFORMATION OF COORDINATES. 



12. To transfer the origin of coordinates to the point (a, ft 7) without 
changing their direction. 

Let it be premised that instead of supposing the coordinate planes at 
right angles to one another, we shall here suppose them to make any 

angles whatever with each other. In this case the axes cease to be rec- 

* 
tangular, but the coordinates x, y, z are still drawn parallel to the axes. 

This being understood, assume 

x = x + , y = / + ft z = z + 7 (39) 

and substitute in the expression involving x, y, z. The result will contain 
x , y , z the coordinates referred to the origin (, ft 7). 

When the substitution is made, the signs of a, ft 7 as explained in (1), 
must be attended to. 

13. To change the direction of the axes from rectangular, without 
affecting the origin. 

Conceive three new axes A x , A y 7 , A z , the first axes being supposed 
rectangular, and these having any given arbitrary direction whatever. 
Take any point ; draw the coordinates x , y , z of this point, and project 
them upon the axis A X. The abscissa x will equal the sum, taken with 
their proper signs, of these three projections, (as is easily seen by drawing 
the figure) ; but if (x x ), (y, y ) ( z > z/ ) denote the angles between the 
axes A x, A x 7 ; A y, A y ; A z, A z respectively ; these projections 
are 

x cos. (x x), y r cos. (y x), z 7 cos. (z! x). 

In like manner we proceed with the other axes, and therefore get 
x x cos. (x x) + y cos. (y x) + ?! cos. (z x) *\ 
y = y cos. (y y) + z cos. (z y) + x cos. (x y) > . . . (40) 
z = z cos. (z z) -f- y cos. (y z) + x cos. (x z) ) 



XVI 



INTRODUCTION. 



Since (x x), (x y), (x z) are the angles which the staight line A x , 
makes with the rectangular axes of x, y, z, we have (5) 
cos. 2 (x x) + cos. s (x y) + cos. 2 x z = 1 ^ 
cos. 2 (y xj 1 + cos. 8 (y y) + cos. 2 (y z) = 1 V ... (41; 
cos. 2 (z x) + cos. 2 (z y) + cos. 2 (z x) = 1 ) 
We also have from (13) p. 



s.(z z) i. 

S.(Z Z) ) 



(42) 



I = cos.(x x)cos.(z x) -f- cos.(x y)cos.(z y ) + cos.(x 
cos.(y z ) ==cos.(y x)cos.(z x) + cos.)y y)cos.(z y)-f-cos.(y z)cos.(z z) 

The equations (40) and (41), contain the nine angles which the axes of 
x , y , z make with the axes of x, y, z. 

Since the equations (41) determine three of these angles only, six of 
them remain arbitrary. Also when the new system is likewise rectangu 
lar, or cos. (x y ) = cos. (x z ) = cos. (y z ) = 1, three others of the 
arbitraries are determined by equations (42). Hence in that case there 
remain but three arbitrary angles. 

14. Required to transform the rectangular axe of coordinates to other 
rectangular axes, having the same origin, but two of which shall be situated 
in a given plane. 

Let the given plane be Y A C, of which the trace in the plane (z, x) is 




Y 



A C. At the distance A C describe the arcs C Y 7 , C x, x x in the planes 
C A Y , (z, x), and X A X. Then if one of the new axes of the coordi 
nates be A X , its position and that of the other two, A Y , A Z , will be 
determined by C x = <p , C x = -4/, and the spherical angle x C x 7 = 6 = 
inclination of the given plane to the plane (z, x). 

Hence to transform the axes, it only remains to express the angles 
(y/x), (y x), &c. which enter the equations (40) in terms of 6 *\> and p. 




ANALYTICAL GEOMETRY. xvii 

By spherics 

cos. (x x) = cos. %}/ cos. + sin. $> sin. cos. 6. 
In like manner forming other spherical triangles, we get 
cos. (y x) = cos. (90 + 0) cos. 4/ + sin. -^ sin. (90 + 0) cos. d 
cos. (x y) = cos. (90 + -^) cos. + sin. (90 + %j/) sin. cos. 6 
cos. (y y) = cos. (90 +^)c 
So that we obtain these four equations 
cos. (x x) = cos. 4* cos. + 

cos. (y ; x) = sin. -v|/ sin. sn. - cos. cos. / . qv 

cos. (x y) r= sin. -\|/ cos. + 
cos. (y y) = sin. ^ sin. + cos - 
Again by spherics, (since A Z is perpendicular to A C, and the inclin 
ation of the planes Z A C, (x, y) is 90 6) we have 

cos (z x) sin. -^ sin. & ~\ . 

cos. (z y) = cos. -^> sin. 6 y ^ 

And by considering that the angle between the planes Z A C, Z A X , = 
90 4- 6, by spherics, we also get 

cos. (x z) =r sin. sin. -\ 

cos. (y z) = cos. sin. ^ v (45) 

cos. (z z) = cos. d } 

which give the nine coefficients of equations (40). 

Equations (41), (42) will also hereby be satisfied when the systems are 
rectangular. 

15. To find the section of a surface made by a plane. 

The last transformation of axes is of great use in determining the na 
ture of the section of a surface, made by a plane, or of the section made 
by any two surfaces, plane or not, provided the section lies in one plane ; 
for having transformed the axes to others, A Z , A X , A Y, the two lat 
ter lying in the plane of the section, by the equations (40), and the de 
terminations of the last article, by putting z = in the equation to the 
surface, we have that of the section at once. It is better, however, to 
make z = in the equations (40), and to seek directly the values of 
cos. (x x), cos. (y x), &c. The equations (40) thus become 

x = x cos. -4> + y sm - 4 cos> 6 ~\ 

y = x sin. -4/ y cos. -vj/ cos. 6 V (46) 

z = y sin. 6 ) 

16. To determine the nature and position of all surfaces of the second 
order : or to discuss the general equation of the second order, viz. 

ax* + by* + cz 2 + 2dxy + 2exz + 2fyz + gx + hy +iz = k . . (a) 
First simplify it by such a transformation of coordinates as shall destroy 



xviii INTRODUCTION. 

the terms in x y, x z, y z ; the axes from rectangular will become oblique, 
by substituting the values (40), and the nine angles which enter these, 
being subjected to the conditions (41), there will remain six of them 
arbitrary, which we may dispose of in an infinity of ways. Equate to 
zero the coefficients of the terms in x y , x z , y z . 

But if it be required that the new axes shall be also rectangular, as this 
condition will be expressed by putting each of the equations (42) equal 
zero, the six arbitrary angles will be reduced to three, which the three 
coefficients to be destroyed will make known, and the problem will thus 
be determined. 

This investigation will be rendered easier by the following process : 
Let x=r a z, y = /3 z be [the equations of the axis of x 7 ; then for 
brevity making 

1 = V (I + a 2 + /3 2 ) 
we find that (3) 

cos. (x x = a 1, cos. (x 7 y) = /S 1, cos. x 7 z = 1. 

Reasoning thus also as to the equations x = a! z, y = $ z of the axis 
of y 7 , and the same for the axis of z , we get 

cos. (y x) = a 7 ! 7 , cos. (y 7 y) = /3 7 1 7 , cos. (y 7 z) = I 7 
cos. (z x) = a 77 1", cos. (z 7 y) = /3" I 77 , cos. (z 7 z) = I 77 . 
Hence by substitution the equations (40) become 
x = 1 a x 7 + I 7 a y 7 + I 77 a 7 
y = l/3x + 
z = 1 x 7 + I 7 y 7 
The nine angles of the problem are replaced by the six unknowns a, 
a 7 , a 77 , /3, /3 7 , (S /7 , provided the equations (41) are thereby also satisfied. 

Substitute therefore these values of x, y, z, in the general equation of- 
the 2d degree, and equate to zero the coefficients of x y 7 , x 7 z , y z 7 , and 
we get 

(a a + d + e) a 77 + (d a + b (3 + f) $" + e a + f + c = > 
(aa 77 + d/3 77 + e) of + (da 7 + b/S 77 + f) /3 7 +e a 77 + f/3 77 + c = J 
One of these equations may be found without the others, and by making 
the substitution only in part. Moreover from the symmetry of the pro 
cess the other two equations may be found from this one. Eliminate a 7 , 
B from the first of them, and the equations x = a! z, y = /3 7 z, of the 
axis of y 7 ; the resulting equation 

(a a + d /3 + e) x + (d a + b /3 + f) y + (e a + f 8 + c] z = . . (b) 
is that of a plane (19). 



I 7 a y + I" a" z -\ 
I 7 & y + 1" /3" z V 
T y + l"z . ) 



ANALYTICAL GEOMETRY. xix 

But the first equation is the condition which destroys the term x y t 
since if we only consider it, a, /?, a , /3 , may be any whatever that will 
satisfy it ; it suffices therefore that the axis of y be traced in the plane 
above alluded to, in order that the transformed equations may not contain 
any term in x y . 

In the same manner eliminating a", jS", from the 2d equation by means 
of the equations of the axis of z , viz. x = a" z, y = /3" z, we shall have 
a plane such, that if we take for the axis of z every straight line which it 
will there trace out, the transformed equation will not contain the term in 
x z\ But, from the form of the two first equations, it is evident that this 
second plane is the same as the first : therefore, if we there trace the axes 
of y and z at pleasure, this plane will be that of y and z , and the 
transformed equation will have no terms involving x y or x z . The 
direction of these axes in the plane being any whatever, we have an in 
finity of systems which will serve this purpose; the equation (b) will be 
that of a plane parallel to the plane which bisects all the parallels to x, 
and which is therefore called the Diametrical Plane. 

Again, if we wish to make the term in y z disappear, the third equa 
tion will give a , @, and there are an infinity of oblique axes which will 
answer the three required conditions. But make x , y , z , rectangular. 
The axis of x must be perpendicular to the plane (y z ) whose equa 
tion we have just found ; and that x = a z, y = /3 z, may be the equa 
tions (see equations b) we must have 

a + d/3 + e = (e + f/3 + c) . . . . (c) 
d a + b + f = (e a + f ,3 + c) /S . . . . (d) 

Substituting in (c) the value of a found from (d) we get 
{(a b)fe + (f 2 eVU/3 3 

+ j (a b) (c b)e+ (2d 2 f 2 e*) e + (2c a b)fd} /3 J 

+ ( (c a) (c b) d+ (2e 2 f 2 d 2 ) d + (2b a c) f e } 

+ (a c) fd + (f 2 d 2 )e = 0. 

This equation of the 3d degree gives for /3 at least one real root ; hence 
the equation (d) gives one for a; so that the axis of x is determined so as 
to be perpendicular to the plane (y , z ,) and to be free from terms in 
x z , and y z . It remains to make in this plane (y, z ,) the axes at right 
angles and such that the term x y may also disappear. But it is evident 
that we shall find at the same time a plane (x , z ,) such that the axis of y 
is perpendicular to it, and also that the terms in x y, z / are not involved. 
But it happens that the conditions for making the axis of y perpendicular 
to this plane are both (c) and (d) so that the same equation of the 3d de- 

62 



xx INTRODUCTION. 

gree must give also P. The same holds good for the axis of z. Conse 
quently the three roots of the equation J3 are all real, and are the values 
of ft /?, |8". Therefore , a , a", are given by the equation (d). Hence, 
There is only one system of rectangular axes which eliminates x y , x z , 
x y ; and there exists wie in all cases. These axes are called the Princi- 
val axes of the Surface. 

Let us analyze the case which the cubic in /3 presents. 

1. If we make 

(a-b)fe + (f 2 e 2 ) d = 

t.he equation is deprived of its first term. This shows that then one of 
the roots of B is infinite, as well as that a derived from equation (d) be 
comes e a + f B = 0. The corresponding angles are right angles. One 
of the axes, that of z for instance, falls upon the plane (x, y), and we 
obtain its equation by eliminating a and {3 from the equations x = a z, 
y = j3 z, which equation is 

ex + fy = 

The directions of y , z are given by the equation in B reduced to a 
quadrature. 

Sndly. If besides this first coefficient the second is also = 0, by substi 
tuting b, found from the above equation, in the factor of [S 2 , it reduces to 
the last term of the equation in ft viz. 

(a c) fd + (f 2 d s ) e = 0. 

These two equations express the condition required. But the coeffi 
cient of 8 is deduced from that of B 2 by changing b into c and d into e, 
and the same holds for the first and last term of the equation in ft 
Therefore the cubic equation is lso thus satisfied. There exists therefore 
an infinity of rectangular systems, which destroy the terms in x y, x z , 
y z. Eliminating a and b from equations (c) and (d) by aid of the above 
two equations of condition, we find that they are the product of fa d 
and e^ d by the common factor eda + fd/3 + fe. These factors 
are therefore nothing ; and eliminating a and ft we find 

fx = dz, ey = d z, e d x + f d y + f e z = 0. 

The two first are the equations of one of the axes. The third that oi 
a plane which is perpendicular to it, and in which are traced the two 
other axes under arbitrary directions. This plane will cut the surface in 
a carve wherein all the rectangular axes are principal, which curve is 
therefore a circle, the only one of curves of the second order which has 
that property. The surface is one then of revolution round the axis 
whose equations we have just given. 



ANALYTICAL GEOMETRY. xxi 

The equation once freed from the three rectangles, becomes of the 
form 

kz 2 + my 2 -fnx 2 + qx + q y-fq"z = h . . . . (e) 

The terms of the first dimension are evidently destroyed by removing 
the origin (39). It is clear this can be effected, except in the cas* 
where one of the squares x 2 , y 2 , z 2 is deficient. We shall examine these 
cases separately. First, let us discuss the equation 

kz 2 + my 2 + nx 2 = h (f) 

Every straight line passing through the origin, cuts the surface in two 
points at equal distances on both sides, since the equation remains the same 
after having changed the signs of x, y, z. The origin being in the middle 
of all the chords drawn through this point is a center. The surface therefore 
has the property of possessing a center whenever the transformed equation 
has the squares of all the variables. 

We shall always take n positive : it remains to examine the cases where 
k and m are both positive, both negative, or of different signs. 

If in the equation (f) k, m, and n, are all positive, h is also positive ; 
and if h is nothing, we have x = 0, y =: 0, z = 0, and the surface has 
but one point. 

But when h is. positive by making x, y, or z, separately equal zero, we 
find the equations to an ellipse, curves which result from the -section of 
the surface in question by the three coordinate planes. Every plane 
parallel to them gives also an ellipse, and it will be easy to show the 
same of all plane sections. Hence the surface is termed an Ellip 
soid. 

The lengths A, B, C, of the three principal axes are obtained by find 
ing the sections of the surface through the axes of x, y, and z. Th^e 
give 

kC 2 = h, mB 2 = h, nA ! = h. 

from which eliminating k, m and n, and substituting in equation (f) we get 

^1-4-^1+ * - - 1 ") 

C* " B 2 "*" A * " I (47) 

A B z 2 + A 2 C 2 y 2 + B 2 C 2 x 2 = A a B* C 2 j 
which is the equation to an Ellipsoid referred to its center and principal 
axes. 

We may conceive this surface to be generated by an ellipse, traced in 
the plane (x, y), moving parallel to itself, whilst its two axes vary, the 
curve sliding along another ellipse, traced in the plane (x, z) as a direct- 

6 3 



xxii INTRODUCTION. 

rix. If two of the quantities A, B, C, are equal, we have an ellipsoid of 
revolution. If all three are equal, we have a sphere. 
Now suppose k negative, and m and h positive or 

k z 2 my 2 ax 2 = h. 

Makings or y equal zero, we perceive that the sections by the planes 
(y z) and (x z), are hyperbolas, whose axis of z is the second axis. All 
planes passing through the axis of z, give this same curve. Hence the 
surface is called an hyperboloid. Sections parallel to the plane (x y) are 
always real ellipses, A, B, C V 1 designating the lengths upon the 
axes from the origin, the equation is the same as the above equation ex 
cepting the first term becoming negative. 
Lastly, when k and h are negative 

kz 2 + my 2 + nx 2 = h; 

all the planes which pass through the axis of z cut the surface in hyper 
bolas, whose axis of z is the first axis. The plane (x y) does not meet 
the surface and its parallels passing through the opposite limits, give 
ellipses. This is a hyperboloid also, but having two vertexes about the 
axis of z. , The equation in A, B, C is still the same as above, excepting 
that the term in z is the only positive one. 
When h = 0, we have, in these two cases, 

k 2 * = my 2 + nx 2 . . . . . . . (48) 

the equation to a cone, which cone is the same to these hyperboloids that 
an asymptote is to hyperbolas. 

It remains to consider the case of k and m being negative. But by a sim 
ple inversion in the axes, this is referred to the two preceding ones. The 
hyperboloid in this case has one or two vertexes about the axis of x ac 
cording as h is negative or positive. 

When the equation (e) is deprived of one of the squares, of x l for in 
stance, by transferring the origin, we may disengage that equation from 
the constant term and from those in y and z ; thus it becomes 

kz 2 + my s = hx (49) 

The sections due to the planes (x z), (x y) are parabolas in the same 
or opposite directions according to the signs of k, m, h ; the planes par 
allel to these give also parabolas. The planes parallel to that of (y z) 
give ellipses or parabolas according to the sign of m. The surface is an 
elliptic paraboloid in the one case, and a hyperbolic paraboloid in the 
other case. When k = m, it is a paraboloid of revolution. 
When h = 0, the equation takes the form 
a * z ~ b y 2 = 



ANALYTICAL GEOMETRY. xxiii 

according to the signs of k and m. In the one case we have 
z = 0, y = 

whatever may be the value of x, and the surface reduces to the axis of x, 
In the other case. 

(a z + b y) (a z by) = 

shows that we make another factor equal zero ; thus we have the system 
of two planes which increase along the axis of x. 

When the equation (e) is deprived of two squares, for instance of x 2 , 
y *, by transferring the origin parallelly to z, we reduce the equation to 

kz 2 + py + qx = (50) 

The sections made by the planes drawn according to the axis of z, are 
parabolas. The plane (x y) and its parallels give straight lines par-r 
allel to them. The surface is, therefore, a cylinder whose base is a para 
bola, or a parabolic cylinder. 

If the three squares in (e) are wanting, it will be that of a plane. 

It is easy to recognise the case where the proposed equation is decom 
posable into rational factors ; the case where it is formed of positive 
squares, which resolve into two equations representing the sector of two 
planes. 



PARTIAL DIFFERENCES. 



17. If u = f (x, y, z, &c.) denote any function of the variable x, y, z, 
&c. d u be taken on the supposition that y, z, &c. are constant, then is the 
result termed the partial difference of u relative to x, and is thus written 

/d u\ , 

( j ) x - 
\d x/ 

Similarly 

rdu, 



( 



denote the partial differences of u relatively to y, z, &c. respectively. 

Now since the total difference of u arises from the increase or decrease 
of its variables, it is evident that 



xxiv INTRODUCTION. 

But, by the general principle laid down in (6) Vol. I, this result may 
be demonstrated as follows ; Let 

u + du = A + Adx+Bdy + C d z +&c. 

A dx 2 + B dy 2 + C dz + &c. | 
+ Mdx.dy+Ndx.dz+&c.J 

Then equating quantities of the same nature, we have 

du = Adx+Bdy+Cdz + &c. 

Hence when d y, d z, &c. = 0, or when y, z, &c. are considered con 
stant 

d u = A d x 
or according to the above notation 



A = 

In the same manner it is shown, that 



&c. 
Hence 



= () d x + (=.) d y + ( ) d z + &c. as before. 



Ex. 1. u =r x y z, &c. 
du\ /du du 

= z 



.. du = yzdx + xzdy + xydz 

du dx dy.dz 
or - = --- \- i + ^ . 
u x y z 

Ex. 2. u = x y z, &c. Here as above 

in = li . . y + ii + & c . 

u x y z 

And in like manner the total difference of any function of any number 
of variables may be found, viz. by first taking the partial differences, as in 
the rules laid down in the Comments upon the first section of the first 
book of the Principia. 

But this is not the only use of partial differences. They are frequently 
used to abbreviate expressions. Thus, in p. 13, and 14, Vol. II. we 



ANALYTICAL GEOMETRY. xxv 

learn that the actions of M, /., p", &c. upon /* resolved parallel to x, 
amount to 

p (x x) y> (x"x) 



(x X )* + (y _y )>+(_ 

*" (x " *) , & MX 

" 



--[(x" x) 2 + (/" y)*-f- (z" z) 2 ]* " [( X + f+ z ) 3 

retaining the notation there adopted. 

But if we make 



V(x-x)* + (y -y) 8 + (z z) 2 = e 

0, 1 

and generally 

V(x"- n X"- m ) 2 + (y"-n_y"...m) 2 + ^ z ...n_ z ...mj 2 Sf 

n, m, 

and then assume 

x = ^ + ^ + & c ...... ,.-,?, (A) 

0, 1 0,2 

+^ + * + &c . ... / ; ; ,,v 



2 1,3 



2, 2,4 

&C. 

we get 

._ W (x x) ^ ^ (x" x) 

~- 



dx 



0, 1 0, 2 



- 

dy 



dz 



0, 1 0,2 



0, 1 0, 2 

We also get 

^ ^ (x x) / d B 



0,1 

dXx ^"( X "_ x ) 



, + 

1, 2 

" x ) AtV (x" x") /dD 



0,3 1,3 2,3 



INTRODUCTION. 

Hence since (B) has one term less than (A) ; (C) one term less than 
) ; and so on ; it is evident that 

Cr") ~*. 



and therefore that 



. 

\dx/ \dx/ T \dx/ \dx" 

See p. 15, Vol. II. 

Hence then X is so assumed that the sum of its partial differences re* 
lative to x, x , x" &c. shall equal zero, and at the same time abbreviate 
the expression for the forces upon p along x from the above complex 
formula into 

d (g + x) IfSl^ , Mx . 

dt 8 ?\&J~ ~JT* 

and the same may be said relatively to the forces resolved parallel to 

y, z, &c. &c. 

Another consequence of this assumption is 



or 



For 

d x N _ w*( x x)y ^"(x" x)y 

s T 



VK x )y . ^VV xQy & __ 

"~ 



. 





3 



&C. 



Hence it is evident that 

t* W x)(y y ) , /.^ (x" x)(y y") &c> 

3 



^V (x x )(y y") ^>"(x "-x) (y-yl &c< 

12 ?2 

^X ( 2 x "_x") (y" y ") M-VCx"" x") (y" f") + &c> 

+ 3 ?* 



23 

&C. 



ANALYTICAL GEOMETRY. xxvii 

In like manner it is found that 

^ (y -y)(x-x) + ^"(y"-v)(x-x-Q &c 

3 3 



y 

0, 1 , 2 

/^> ; (y" y ) (x x") i*i*"(y" y) (* x ") 
+ ~p- + &c. 

1, 2 1, 3 

&c. 

which is also perceptible from the substitution in the above equation of 
y for x, x for y ; y for X , x for y ; and so on. 

But 

(y -y) ( x x ) = ( x x ) (y y ) 

(y" y) (x x") = (x" x) (y y") 

&c. 
consequently 

2 x 

c 

See p. 16. For similar uses of partial differences, see also pp. 22, and 
105. 



CHANGE OF THE INDEPENDENT VARIABLE. 

When an expression is given containing differential coefficients, sucli 
as 

dj d 2 y 
ci x d x 

in which the first differential only of x and its powers are to be found, it 
shows that the differential had been taken on the supposition that dx is 
constant, or that d 2 x = 0, d x = 0, and so on. But it may be re 
quired to transform this expression to another in which d*x, d 3 x shall 
appear, and in which d y shall be constant, or from which d 2 y, &c. shall 
be excluded. This is performed as follows : 

For instance if we have the expression 

dy 2 



1 + 



d x 1 d y 



dx* 
the differential coefficients 

d y d^y 
d x dx" 



xxviii INTRODUCTION. 

may be eliminated by means of the equation of the curve to which we 
mean to apply that expression. For instance, from the equation to a 
parabola y = a x 2 , we derive the values of 

dy . d 2 y 
-j-^- and -T-i 
d x dx 2 

which being substituted in the above formula, these differential coefficients 
will disappear. If we consider 

dy d* y 
dlE die 2 

unknown, we must in general have two equations to eliminate them from 
one formula, and these equations will be given by twice differentiating the 
equation to the curve. 

When by algebriacal operations, d x ceases to be placed underneath 
d y, as in this form 

. _ y(dx + dy 2 ) (52) 

d x * + dy 2 y dy 

the substitution is effected by regarding d x, d y, d z y as unknown; but 
then in order to eliminate them, there must be in general the same 
number of equations as of unknowns, and consequently it would seem the 
elimination cannot be accomplished, because by means of the equation to 
the curve, only two of the equations between d x, d y, d 2 y can be ob 
tained. It must be remarked, however, that when by means of these two 
equations we shall have eliminated d y and d 2 y, there will remain a com 
mon factor d x 8 , which will also vanish. For example, if the curve is 
always a parabola represented by the equation y =. ax , by differentiat 
ing twice we obtain 

dy = 2axdxOd 2 y = 2a dx* 

and these being substituted in the formula immediately above, we shall 
obtain, after suppressing the common factor d x 2 , 



4 a 2 x 2 Say 

The reason why d x 2 becomes a common factor is perceptible at once, 
for when from a formula which primitively contained 

d y dy 
d x 2 d x 

we have taken away the denominator of pJ all the terms independent 

of ^-2- and V^ must acquire the factor d x 2 ; then the terms which 
d x 2 d x 

were affected by -r-^ do not contain dx, whilst those affected by j* 



ANALYTICAL GEOMETRY. xxix 

contain d x. When we afterwards differentiate the equation of the curve, 
and obtain results of the form dy = M d x, d 2 y = Ndx 2 , these values 
being substituted in the terms in d 2 y, and in dy dx, will change them, 
as likewise the other terms, into products of d x 2 . 

What has been said of a formula containing differentials of the two first 
orders applying equally to those in which these differentials rise to supe 
rior orders, it thence follows that by differentiating the equation of the 
curve as often as is necessary, we can always make disappear from the 
expression proposed, the differentials therein contained. 

The same will also hold good if, beside these differentials which we have 
just been considering, the formula contain terms in d x, in d 3 x, &c. ; 
for suppose that there enter the formula these differentials d x, d y, d " x, 
d 2 y and that by twice differentiating the equation represented by y = f x, 
we obtain these equations 

F (x, y, d y, d x) = 
F(x,y,dx,dy,dx,dy) = 0, 

we can only find two of the three differentials d y, d 2 x, d ~ y, and we see 
it will be impossible to eliminate all the differentials of the formula ; there 
is therefore a condition tacitly expressed by the differential d 2 x; it is 
that the variable x is itself considered a function of a third variable which 
does not enter the formula, and which we call the independent variable. 
This will become manifest if we observe, that the equation y = f x may 
be derived from the system of two equations 

x =: F t, y = p t 
from which we may eliminate t. Thus the equation 

(x c) s 
v n J 

y b * 

is derived from the system of two equations 

x = b t + c, y = a t 2 , 

and we see that x and y must vary by virtue of the variation which t may 
undergo. But this hypothesis that x and y vary as t alters, supposes that 
there are relations between x and t, and between y and t. One of these 
relations is arbitrary, for the equation which we represent generally by 
y = f x, for example 

a / \ . 

y = b - (x c) *, 

if we substitute between x and t, the arbitrary relation, 

t 3 
x ~ ~ 



xxx 



INTRODUCTION. 



this value being put in the equation 
will change it to 



y = ( x 



an equation which, being combined with this, 



ought to reproduce by elimination, 

(x c) 

y = a^ 5 - r -, 

the only condition which we ought to regard in the selection of the varia 
ble t. 

We may therefore determine the independent variable t at pleasure. 
For example, we may take the chord, the arc, the abscissa or ordinate 
for this independent variable ; if t represent the arc of the curve, we 
have 

t = V (dx + dy 2 ); 

if t denote the chord and the origin be at the vertex of the curve, we 
have 

t = V (x 2 + y 2 ); 
lastly, if t be the abscissa or ordinate of the curve, we shall have 

t = x, or t = y. 

The choice of one of the three hypotheses or of any other, becoming in- 
dispensible in order that the formula which contains the differentials, may 
be delivered from them, if we do not always adopt it, it is even then tacitly 
supposed that the independent variable has been determined. For ex 
ample, in the usual case where a formula contains only the differentials 
d x, d y, d 2 y, d 3 y, &c. the hypothesis is that the independent variable 
t has been taken for the abscissa, for then it results that 

dx 

i = x 31 =: 1} 

d 2 -K 

5-4 = 0, 

d t 2 

4^1 = 0, &c. 

d t - 

and we see that the formula does not contain ths second, third, &c. dif 
ferentials. 



ANALYTICAL GEOMETRY. xxxi 

To establish this formula, in all its generality, we must, as above, sup 
pose x and y to be functions of a third variable t, and then we have 

d y _ d y d x 
dT "" cTx* dT 
from which we get 

ai = ini (53) 

ft 

taking the second differential of y and operating upon the second meinbei 
as if a fraction, we shall get 

d x d * y d y d 8 x 
d 8 y _ d~t cU dT d t 
dx " " dx 2 

d t 2 

In this expression, d t has two uses; the one is to indicate that it is 
the independent variable, and the other to enter as a sign of algebra. 
In the second relation only will it be considered, if we keep in view that 
t is the independent variable. Then supposing d t 2 the common factor, 
the above expression simplifies into 

d 2 y _ dxd 2 y dy d 2 x 
dx = d x 8 

and dividing by d x, it will become 

d* y __ d x d* y dyd 2 x 
die 2 = dx 3 

Operating in the same way upon the equation (53), we see that in 
taking t as the independent variable, the second member of the equation 
ought to become identical with the first ; consequently we have only one 
change to make in the formula which contains the differential coefficients 

d y d 2 y d 2 v 

-j ~T~i > V1Z - to replace J t by 

dxd z y dyd 2 x 

d x 2 * V / 

To apply these considerations to the radius of curvature which is given 
by the equation See p. 61. vol. I.) 

i 



dx 



xxxii INTRODUCTION. 

if we wish to have the value of R, in the case where t shall be the inde 
pendent variable, we must change the equation to 

n (!*&) . ;; ;. 

dx d 2 y d y d - x * 

dx 3 
and observing that the numerator amounts to 

(dx + dy )* 

dx 3 
we shall have 

dy f 



R - 



cTx^d y dy d x 
This value of R supposes therefore that x and y are functions of a third 
independent variable. But if x be considered this variable, that is to say, 
if t = x, we shall have d 2 x =0, and the expression again reverts to the 
common one 



(dx +dy*)* V 1 + dx ) 

dxd y d 2 y 



dx 

But if, instead of x for the independent variable, we wish to have the 
ordinate y, this condition is expressed by y = t ; and differentiating this 
equation twice, we have 



The first of these equations merely indicates that y is the independent 
variable, which effects no change in the formula ; but the second shows 
us that d * y ought to be zero, and then the equation (55) becomes 

_(dx + dy)* (56) 

dy d 2 x 

We next remark, that when x is the independent variable, and 
consequently d 2 x = 0, this equation indicates that d x is constant. 
Whence it follows, that generally the independent variable has always 
a constant differential. 

Lastly, if we take the arc for the independent variable, we shall have 

dt = V (dx 2 + dy 1 ); 
Hence, we easily deduce 

dx* , d_y 2 .. 
+ 



ANALYTICAL GEOMETRY. xxxiii 

differentiating this equation, we shall regard d t as constant, since t is the 
independent variable ; we get 

2 d xd x 2dy d 2 y 
~d~F~ dt " 

which gives 

dxd 2 x = d y d 2 y 

Consequently, if we substitute the value of d * x, or that of d z y, in the 
equation (55), we shall have in the first case 

. V(dx* + dy ) 

II A 15 il A . I <1 I I 





(d x + d y 2 ) d " y d * y 

and in the second case, 

j 
(dx 2 -f dy 2 ) 2 , V (dx 2 + dy 2 ) , .,... 

li = 7-5 , J , j d y = > n J L d y . (58) 

(d x + d y ) d x J d x 

In what precedes, we have only considered the two differential coeffi 
cients 

( Ii ily . 

but if the formula contain coefficients of a higher order, we must, by 
means analogous to those here used, determine the values of 

^Xf^ &c 

do vl I jj CVV. 

x J d x * 

which will belong to the case where x and y are functions of a third in 
dependent variable. 



PROPERTIES OF HOMOGENEOUS FUNCTIONS. 

IfMdx + Ndy -f Pdt + &o. = d z, be a homogeneous function of 
any number of variables, x, y, t, &c. in which the dimension of each term is 
n, then is 

MX + Ny + Pt + &c. = nz. 

For let M d x + N d y be the differential of a homogeneous function 
z between two variables x and y ; if we represent by n the sum of the 
exponents of the variables, in one of the terms which compose this func 
tion, we shall have therefore the equation 
Mdx + Ndy = dz. 

y 

Making * = q, we shall find (vol. I.) 
F(q) X x" = z; 



xxxiv INTRODUCTION. 

and replacing, in the above equation, y by its value q x, and calling M 
N , what M and N then become, that equation transforms to 

M d x + N d. q x = d z ; 
and substituting the value of z, we shall have 

M d x -f N d (q z) = d (x " F. q.) 
But d (q z) =: q d x + x d q. Therefore 

(M + N q) dx + N xdq = d (x n F. q). 

But, (M + N q) d x being the differential of x " F q relatively to x, we 
have (Art. 6. vol. 1.) 

M + N q = nx"- 1 X F. q. 
If in this equation y be put for q x, it will become 

M + N- = nx - F. q, 
x 

or 

Mx+Ny = nz. 

This theorem is applicable to homogeneous functions of any number of 
variables ; for if we have, for example, the equation 
M d x + Ndy+ Pdtrrdz, 
in which the dimension is n in every term, it will suffice to make 

y t 

= q> = r 

x ^ x 

to prove, by reasoning analogous to the above, that we get z x" F (q, r), 
and, consequently, that 

Mx + Ny+Pt = nz (59) 

and so on for more variables. 



THEORY OF ARBITRARY CONSTANTS. 

An equation V = between x, y, and constants, may be considered as 
the complete integral of a certain differential equation, of which the order 
depends on the number of constants contained in V = 0. These constants 
are named arbitrary constants, because if one of them is represented by a, 
and V or one of its differentials is put under the form f (x, y) = a, we see 
that a will be nothing else than the arbitrary constant given by the integra 
tion of d f (x, y). Hence, if the differential equation in question is of the 
order n, each integration introducing an arbitrary constant, we have 
V == 0, which is given by the last of three integrations, and contains, at 



ANALYTICAL GEOMETRY. xxxv 

least, n arbitrary constants more than the given differential equation. Let 
therefore 



F(x,y) = 0,F x,y, = 0,F x , y ,, = &c. (a) 



be the primitive equation of a differential equation of the second order 
and its immediate differentials. 

Hence we may eliminate from the two first of these three equations, 
the constants a and b, and obtain j 



If, without knowing F (x, y) = 0, we find these equations, it will be 

sufficient to eliminate from them -r* , to obtain F (x, y) = 0, which will 

dx 

be the complete integral, since it will contain the arbitrary constants a, b. 
If, on the contrary, we eliminate these two constants between the 
above three equations, we shall arrive at an equation which, containing 
the same differential coefficients, may be denoted by 

d d 2 



But each of the equations (b) will give the same. In fact, by eliminating 
the constant contained in one of these equations and its immediate differ 
ential, we shall obtain separately two equations of the second order, 
which do not differ from equation (c) otherwise than the values of x and 
of y are not the same in both. Hence it follows, that a differential equa 
tion of the second order may result from two equations of the first order 
which are necessarily different, since the arbitrary constant of the one is 
different from that of the other. The equations (b) are what we call the 
first integrals of the equation (c), which is independent, and the equation 
F (x, y) = is the second integral of it. 

Take, for example, the equation y = a x + b, which, because of its 
two constants, rnay be regarded as the primitive equation of an equation 
of the second order. Hence, by differentiation, and then by elimination 
of a, we get 

d y d y . i 

-5-^- = a , y = x -r -f b. 
dx dx 

These two first integrals of the equation of the second order which we 
are seeking, being differentiated each in particular, conduct equally, by 

1 2 

the elimination of a, b, to the independent equation -. ^ = 0. In the 

c3 



xxxvi INTRODUCTION. 

case where the number of constants exceeds that of the required arbitrary 
constants, the surplus constants, being connected with the same equations, 
do not acquire any new relation. Required, for instance, the equation of 
the second order, whose primitive is 



differentiating we get 

iZ = ax + b. 
dx 

The elimination of a, and then that of b, from these equations, give 
separately these two first integrals 

5-Z = ax + b, y = X C |^ \ ax 2 + c . . . (d) 
dx J dx 

Combining them each with their immediate differentials, we arrive, 

d 2 y 
by two different ways, at -, - = a. If, on the contrary, we had elimi 

nated the third constant a between the primitive equation and its imme 
diate differential, that would not have produced a different result; for 
we should have arrived at the same result as that which would lead to 
the elimination of a from the equations (d), and we should then have 

1 2 *J 

fallen upon the equation x -j-f z = -^ b, an equation which reduces 

d 2 y 

to -j ~ = a by combining it with the first of the equations (d). 

Let us apply these considerations to a differential equation of the third 
order : differentiating three times successively the equation F (x, y) = 0, 
we shall have 

F (x,y, to = 0, F(x,y, d ^, ^) = 0, F (x, y,-^,^ , ^ = 
V >t7 dx/ V Jy dx dxV \ J dx dx* dx 3 / 

These equations admitting the same values for each of the arbitrary 
constants contained by F (x, y) z= 0, we may generally eliminate these 
constants between this latter equation and the three preceding ones, and 
obtain a result which we shall denote by 

c I dy d 2 y d 3 y\ 

f ( x y>si d/"d^) = ...... w 

This will be the differential equation of the third order of F (x, y) = 0. 
and whose three arbitrary constants are eliminated ; reciprocally, 
F (x, y) =r 0, will be the third integral of the equation (e). 

If we eliminate successively each of the arbitrary constants from the 



ANALYTICAL GEOMETRY. xxxvii 

equation F (x, y) = 0, and that which we have derived by differentiation, 
we shall obtain three equations of the first order, which will be the "second 
integrals of the equation (e). 

Finally, if we eliminate two of the three arbitrary constants by means 
of the equation F (x, y) = 0, and the equations which we deduce by two 
successive differentiations, that is to say, if we eliminate these constants 
from the equations 

F (x, y ) = 0, F (*,y, ) = 0, F (x, y , 1|, ) = . . (f) 

we shall get, successively, in the equation which arises from the elimina 
tion, one of the three arbitrary constants ; consequently, we shall have as 
many equations as arbitrary constants. Let a, b, c, be these arbitrary 
constants. Then the equations in question, considered only with regard 
to the arbitrary constants which they contain, may be represented by 

p c = 0, <p b = 0, <f> a = (g) 

Since the equations (f) all aid in the elimination which gives us one of 
these last equations, it thence follows that the equations (g) will each be 
of the second order; we call them the first integrals of the equation (e). 

Generally, a differential equation of an order n will have a number n 
of first integrals, which will contain therefore the differential coefficients 

from -T* to , D _/ t inclusively; that is to say, a number n _ 1 of differential 

( 1 X. (I X 

coefficients ; and we see that then, when these equations are all known, 
to obtain the primitive equation it will suffice to eliminate from these equa 
tions the several differential coefficients. 



PARTICULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS. 

It is easily seen that a particular integral may always be deduced from 
the complete integral, by giving a suitable value to the arbitrary con 
stant. 

For example, if we have given the equation 

xdx + ydy = dyVx* + y 2 a*, 
whose complete integral is 

y + c = V (x 2 + y 2 a ), 
whence (for convenience, by rationalizing,) we get 



c2 



xxxviii INTRODUCTION. 

and the complete integral becomes 

2 cy + c 2 x 2 + a 2 = . . . . (i) 

Hence, in taking for c an arbitrary constant value c = 2 a, we shall 
obtain this particular integral 

2 cy + 5 a 2 x 2 = 0, 

which will have the property of satisfying the proposed equation (h) as 
well also as the complete integral. In fact, we shall derive from this 
particular integral 

x 2 5 a g cl y x_ 

~2~c d x =: "c 
these values reduce the proposed to 



an equation which becomes identical, by substituting in the second mem 
ber, the value of c 2 , which gives the relation c = 2 a. Let 

Mdx + Ndy = 0, 

be a differential equation of the first order of a function of two variables 
x and y ; we may conceive this equation as derived by the elimination of 
a constant c from a certain equation of the same order, which we shall 
represent by 

m d x + n d y = 0, 
and the complete integral 

F (x, y, c) = 0, 

which we shall designate by u. But, since every thing is reduced to 
taking the constant c such, that the equation 
Mdx + Ndy = 0, 

may be the result of elimination, we perceive that is at the same time 
permitted to vary the constant c, provided the equation 

Mdx + Ndy = 0, 
holds good ; in this case, the complete integral 

F (x, y, c) = 

will take a greater generality, and will represent an infinity of curves of 
the same kind, differing from one another by a parameter, that is, by a 
constant. 

Suppose therefore that the complete integral being differentiated, by 
considering c as the variable, we have obtained 

< y = (ai) " * + (il) de 



ANALYTICAL GEOMETRY. xxxix 

an equation which, for brevity, we shall write 

d y = p d x + q d c (k) 

Hence it is clear, that if p remaining finite, q d c is nothing, the result 
of the elimination of c as a variable from 

F (x, y, c) = 0, 

and the equation (k), will be the same as that arising from c considered 
constant, from 

F (x, y, c) = 0, 
and the equation 

d y = p d x 
(this result is on the hypothesis 

Mdx+Ndy = 0), 
for the equation (k), since 

q d c = 0, 
does not differ from 

dy = p d x; 

but in order to have 

q d c = 0, 

one of the factors of this equation -sz constant, that is to say, that we 
have 

d c = 0, or q =. 0. 

In the first case, d c =r 0, gives c = constant, since that takes place 
for particular integrals ; the second case, only therefore conducts to a par 
ticular solution. But, q being the coefficient of d c of the equation (k), 
we see that q = 0, gives 

dx " 

This equation will contain c or be independent of it. If it contain c, 
there will be two cases ; either the equation q = 0, will contain only c 
and constants, or this equation will contain c with variables. In the first 
case, the equation q = 0, will still give c = constant, and in the second case, 
it will give c = f (x, y) ; this value being substituted in the equation 
F (x, y, c) = 0, will change it into another function of x, y, which will 
satisfy the proposed, without being comprised in its complete integral, 
and consequently will be a singular solution ; but we shall have a parti 
cular integral if the equation c = f (x, y), by means of the complete "n- 
tegral, is reduced to a constant. 

c4 



xl INTRODUCTION. 

When the factor q = from the equation q d c = not containing 
the arbitrary constant c, we shall perceive whether the equation q =: 
gives rise to a particular solution, by combining this equation with the 
complete integral. For example, if from q = 0, we get x = M, and put 
this value in the complete integral F (x, y, c) = 0, we shall obtain 
c = constant = B or c = fy; 

In the first case, q = 0, gives a particular integral j for by changing c 
into B in the complete integral, we only give a particular value to the 
constant, which is the same as when we pass from the complete integral 
to a particular integral. In the second case, on the contrary, the value 
f y introduced instead of c in the complete integral, will establish between 
x and y a relation different from that which was found by merely re 
placing c by an arbitrary constant. In this case, therefore, we shall have 
a particular solution. What has been said of y, applies equally to x. 

It happens sometimes that the value of c presents itself under the form 

: this indicates a factor common to the equations u and U which is ex 

traneous to them, and which must be made to disappear. 

Let us apply this theory to the research of particular solutions, when 
the complete integral is given. 

Let the equation be 

y dx xdy = a V^dx 2 -f- dy ! ) 
of which the complete integral is thus found. 

Dividing the equation by d x, and making 



we obtain 

y px = a V(l + p*). 

Then differentiating relatively to x and to p, we get 

, . a p d p 

dy pdx xdp = V(1 f +pt) ; 

observing that 

dy = pdx, 

this equation reduces to 

, a p d p A 

p - 



and this is satisfied by making d p = 0. This hypothesis gives p = con 
stant s= c, a value which being put in the above equation gives 



ANALYTICAL GEOMETRY. xli 

y ex = a V(\ + c ) (!) 

This equation containing an arbitrary constant c, which is not to be 
found in the proposed equation, is the complete integral of it. 

This being accomplished, the part q d c of the equation d y = p d x + 
q d c will be obtained by differentiating the last equation relatively to c 
regarded as the only variable. Operating thus we shall have 

, a c d c 

xdc + =0; 



consequently the coefficients of d c, equated to zero, will give us 

ac 



x = 



he 2 ) 
To find the value of c, we have 



a r 2 

il V | 



which gives 



and 



- V(a 2 x 2 ) 

by means of this last equation, eliminating the radical of the equation (m) 
we shall thus obtain 



c = 



This value and that of V (I + c 2 ) being substituted in the equation (D 
will give us 

x 2 a 2 

V(a 2 x 2 ) = : V(a 2 x z ) 
whence is derived 

y = V(a ! x 2 ), 

an equation which, being squared, will give us 

y 2 = a* x 2 ; 

and we see that this equation is a particular solution, for by differentiating 
it we obtain 

x d x 
d y = -- ; 

y 

this value and that of V(x 2 + y = ), being substituted in the equation 
originally proposed, reduce it to . 

a 2 = a 8 . 
In the application which we have just given, we have determined the 



xbi INTRODUCTION. 

/d y\ 
value of c by equating to zero the differential coefficient (-r^J. This 

process may sometimes prove insufficient. In fact, the equation 

being put under this form 

Adx + Bdy + Cdc = 
where A, B, C, are functions of x and y, we shall thence obtain 

d - _ dx - dc (o) 

B 1 C J 
d x = -jr-d y ir" c (P) 

and we perceive that if all that has been said of y considered a function of 
x, is applied to x considered a function of y, the value of the coefficient of 
d c will not be the same, and that it will suffice merely that any factor of B 
destroys in C another factor than that which may destroy a factor of A, 
in order that the value of the coefficient of d c, on both hypotheses, may 
appear entirely different. Thus although very often the equations 



give for c the same value, that will not always happen ; the reason of 
which is, that when we shall have determined c by means of the equation 

^=0, 
dc 

d x 

it will not be useless to see whether the hypothesis of -= gives the same 

result. 

Clairaut was the first to remark a general class of equations susceptible 
of a particular solution ; these equations are contained in the form 

dy .p, dy 

y = -f^-x + F. -j^- 

d x f d x 

an equation which we shall represent by 

y = px + Fp ......... (r) 

By differentiating it, we shall find 



tins equation, since d y = p d x, becomes 



ANALYTICAL GEOMETRY. xliii 

and since d p is a common factor, it may be thus written : 



We satisfy this equation by making d p = 0, which gives p = const. 
= c; consequently, by substituting this value in the equation (r) we 
shall find 

y = ex + F c . 

This equation is the complete integral of the equation proposed, since 
an arbitrary constant c has been introduced by integration. If we differ 
entiate relatively to c we shall get 

Fc x ) , 

- tlc - 



Consequently, by equating to zero the coefficients of d c, we have 

d Fc 

* + ^ = > 

which being substituted in the complete integral, will give the particular 
solution. 



THE INTEGRATION OF EQUATIONS OF PARTIAL DIFFERENCES. 



An equation which subsists between the differential coefficients, com 
bined with variables and constants, is, in general, a partial differential 
equation, or an equation of partial differences. These equations are thus 
named, because the notation of the differential coefficients which they 
contain indicates that the differentiation can only be effected partially ; 
that is to say, by regarding certain variables as constant. This supposes, 
therefore, that the function proposed contains only one variable. 

The first equation which we shall integrate is this ; viz. 

/d z\ 

(dx) =a 

If contrary to the hypothesis, z instead of being a function of two vari 
ables x, y, contains only x, we shall have an ordinary differential equation, 
which, being integrated, will give 

z = a x + c 

but, in the present case, z being a function of x and of y, the ys con 
tained in z have been made to disappear by differentiation, since differen- 



xhv INTRODUCTION. 

tiating relatively to x, we have considered y as constant. We ought, 
therefore, when integrating, to preserve the same hypothesis, and suppose 
that the arbitrary constant is m general a function of y ; consequently, we 
shall have for the integral of the proposed equation 

z = ax + py. 
Required to integrate the equation 



. 

in which X is any function of x. Multiplying by d x, and integrating, 

we get 

z =/Xdx + py. 

For example, if the function X were x 2 + a 2 , the integral would be 

z = ^ + a 2 x + ?y. 
In like manner, it is found that the integral of 



is 

z = x Y + <p y . 

Similarly, we shall integrate every equation in which (:rr) is equal to 

a function of two variables x, y. If, for example, 
/d zx _ x 

Vd x/ " V a y + x 2 

considering y as constant, we integrate by the ordinary rules, making the 
arbitrary constant a function of y. This gives 

z = V (ay + x 2 ) + py. 
Finally, if we wish to integrate the equation 



V(y 2 x 2 ) 
regarding y as constant, we get 

i x , 
z = sm.~ l --- f- 9 y 

*/ 

Generally to integrate the equation 



we shall take the integral relatively to x, and adding to it an arbitrary 
function of y, as the constant, to complete it, we shall find 
z = /T(x, y) dx + ty. 



ANALYTICAL GEOMETRY. xlv 

Now let us consider the equations of partial differences which contain 
two differential coefficients of the first order ; and let the equation be 



in which M and N represent given functions of x, y. Hence 

M 



substituting this value in the formula 

dz = 



o 

fd 



which has no other meaning than to express the condition that z is a 
function of x and of y, we obtain 



/d z\ ( , M 

tlz = (dx) | dx - -N 



or 



/d z\ Ndx Mdy 
d z = ( -j ) - - 
Vdx/ 



N 

Let X be the factor proper to make Ndx Mdya complete differ 
ential d s ; we shall have 

X (N d x M d v) = d s. 

By means of this equation, we shall eliminate Ndx Mdy from the 
preceding equation, and we shall obtain 

, 1 /d z\ , 

d z = - XT . (T ). d s. 
X N \dx/ 

Finally, if we remark that the value of ( C j ) is indeterminate, we may 

take it such that ^-^ . ( ^ \ d s may be integrable, which would make it 

a function of s ; for we know that the differential of every given function 
of s must be of the form F s . d s. It therefore follows, that we may 



assume 

z 



an equation which will change the preceding one into 

d z = F s . d s 
which gives 

z = 9 s. 



xlvi INTRODUCTION. 

Integrating by this method the equation 

/d z\ /d z\ 

X [T 1 V (-] ) = 
\dy/ . J VI x/ 

we have in this case 

M = - y, 
and 

N = x; 
consequently 

d s = X (x d x + y d y). 
It is evident that the factor necessary to make this integrable is z. 

Substituting this for X and integrating, we get 

s = x 2 + y z . 
Hence the integral of the proposed equation is 

z = f (x= + y 2 ). 
Now let us consider the equation 



. 

in which P, Q, R are functions of the variables x, y, z ; dividing it by P 
and making 

Q -M -5-N 
p _ ivi, p . 

we shall put it under this form : 

+N = 0; 



y 

and again making 



and 



it becomes 

p + M q + N = ........... (a) 

This equation establishes a relation between the coefficients p and q of 
the general formula 

d 



= pdx + qdy; 

without which relation p and q would be perfectly arbitrary, for as it has 
been already observed, this formula has no other meaning than to indicate 
that z is a function of two variables x, y, and that function may be any 



ANALYTICAL GEOMETRY. xlvii 

whatever ; so that we ought to regard p and q as indeterminate m ihis last 
equation. Eliminating p from it, we shall obtain 
dz + Ndx = q(dy-Mdx) 

and q will remain always indeterminate. Hence the two members of this 
equation are heterogeneous (See Art. 6. vol. 1), and consequently 

dz + Ndx = 0, dy M d x = ..... (b) 

If P, Q, R do not contain the variable z, it will be the same of M and 

N ; so that the second of these equations will be an equation of two varia 

bles x and y, and may become a complete differential by means of a factor 

A. This gives 

X (d y M d x) = 0. 

The integral of this equation will be a function of x and of y, to wluca 
we must add an arbitrary constant s ; so that we shall have 

F ( x > y) = s; 
whence we derive 

y = f (x, s). 

Such will be the value of y given us by the second of the above equa 
tions; and to show that they subsist simultaneously we must substitute 
this value in the first of them. But although the variable y is not shown, 
it is contained in N. This substitution of the value of y just found, 
amounts to considering y in the first equation as a function of x and of 
the arbitrary constant s. Integrating therefore this first equation on that 
hypothesis we find 

z = yN d x + <p s. 
To give an example of this integration, take the equation 



and comparing it with the general equation (a), we have 

M = 2- , N = V (x 2 + y 2 ). 
x x J 

These values being substituted in the equations (b) will change them to 



d z V (x* + y 2 ) d x = 0, d y 2- d x = 

X X 



Let A be the factor necessary to make the last of these integrable, and 
we have 

x(dy-I-dx) = 0, 
or rather 



xlviii INTRODUCTION. 

1 V 

which is integrable when X = ; for then the integral is -J = constant. 

Put therefore 

Z=s 

X 

and consequently 

y = s x. 

By means of this value of y, we change the first of the equation!* 
(c) into 



or rather into 



* x sx , 
d z a - . d x = 0, 



Integrating on the supposition that s is constant, we. shall obtain 

z = a/dx V (1 + s 2 ) + <p s 
and consequently 

z = a x V (1 + s 2 ) + <p s. 
Substituting for s its value we get 



= a 



In the more general case where the coefficients P, Q, R of the equation 
contain the three variables x, y, z it may happen that the equations 
(.b) contain only the variables which are visible, and which consequently 
we may put under the forms 

d z = f (x, z) d x = 0, d y = F (x, y) d x. 

These equations may be treated distinctly, by writing them as above, 

z =/f(x,z)dx + z, y =/F (x,y) dx + <Dy 
for then we see we may make z constant in the first equation and y in 
the second ; contradictory hypotheses, since one of three coordinates 
x, y, z cannot be supposed constant in the first equation without its being 
not constant in the second. 

Let us now see in what way the equations (b) may be integrated in the 
case where they only contain the variables which are seen in them. 

Let p and X be the factors which make the equations (b) integrable. 
If their integrals thus obtained be denoted by U and by V, we have 
A (d z + N d x) = d U, A (d y M d x) - d V. 



ANALYTICAL GEOMETRY. xlix 

By means of these values the above equation will become 

dU = q-dV . ... .-. . . (d). 

Since the first member of this equation is a complete differential the 
second is also a complete differential, which requires q to be a function 

of V. Represent this function by <f> V. Then the equation (d) will 
become 

dU = pV.dV 
which gives, by integrating, 

U = <i>V. 
Take, for example, the equation 

/d z\ . (^-\ 

Xjr \dx/ \dy/ "" 

which being written thus, viz. 

X/f\ "7 rr 
fV ^\ n 

-i I =r U 



we compare it with the equation 



and obtain 



M = X -, N = - 

y x 



By means of these values the equations (b) becomes 

dz -. dx = 0,dy ~dx = 0; 
x y 

which reduce to 

xdz zdx = 0,ydy xdx = 0. 

The factors necessary to make these integrable are evidently ^ and 2. 

JH 
f 

Substituting which and integrating, we find and y 2 x 2 for the in- 

X 

tegrals. Putting, therefore, these values for U and V in the equation 
U = * V, we shall obtain, for the integral of the proposed equation, 

- = cD (y 2 _ x 2 ) 
X 

It must be remarked, that, if we had eliminated q instead of p, the equa 
tions (b) would have been replaced by these 

Mdz + Ndy=0,dy Mdx = 0. . . . (e) 
and since all that has been said of equations (b) applies equally to these, 

d 



I INTRODUCTION. 

it follows that, in the case where the first of equations (b) was not in- 
tegrable, we may replace those equations by the system of equations (e), 
which amounts to employing the first of the equations (e) instead of the 
first of the equations (b). 
For instance, if we had 

/d z\ /d z\ 

this equation being divided by a z and compared with 

will give us 

a a z 

and the equations (b) will become 

XV X 

d z H *- dx = 0,dy H dx = 0; 

r a z a 

which reduce to 

azdz + xydx=rO,ady + xdx = . (0 
The first of these equations, which, containing three variables, is not 
immediately integrable, we replace by the first of the equations (e), and 
we shall have, instead of the equations (f), these 

d z + d y = 0,ady + xdx = 0; 

a a z J 

which reduce to 

2ydy 2zdz = 0,2ady + 2xdx = 0; 
equations, whose integrals are 

y 2 z 8 , 2ay + x 2 - 

These values being substituted for U and V, will give us 
y 2 z 1 = <p (2ay + x 2 ). 

It may be remarked, that the first of equations (e) is nothing else than 
the result of the elimination of d x from the equations (b) . 

Generally we may eliminate every variable contained in the coefficients 
M, N, and in a word, combine these equations after any manner what 
ever ; if after having performed these operations, and we obtain two in 
tegrals, represented by U = a, V = b, a and b being arbitrary constants, 
we can always conclude that the integral is U = * V. In fact, since 
a and b are two arbitrary constants, having laken b at pleasure, we may 
compose a in terms of b in any way whatsoever ; which is tantamount to 
saying that we may take for a an arbitrary function of b. This condition 
will be expressed by the equations a = <p (b). Consequently, we shall 



ANALYTICAL GEOMETRY. 11 

have the equations U = <p b, V = b, in which x, y, z represent the same 
coordinates. If we eliminate (b) from these equations, we shall obtain 
U = pV. 

This equation also shows us that in making V = b, we ought to have 
U = f b = constant ; that is to say, that U and V are at the sanie time 
constant; without which a and b would depend upon one another, where 
as the function p is arbitrary. But this is precisely the condition expressed 
by the equations U = a, V = b. 

To give an application of this theorem, let ^ 

d z\ /d 



Dividing by z x and comparing it with the general equation we 
have 

M = , N = ?; 



ZX 



and the equations (b) give us 



dz dx = 0,dy-fdx = 

zx J r x 

or 

zxdz y * d x = 0, xdy + ydz=0. 

The first of these equations containing three variables we shall not at 
tempt its integration in that state ; but if we substitute in it for y d x its 
value derived from the second equation, it will acquire a common factor 
x, which being suppressed, the equation becomes 

z d z + y d y = 0, 

and we perceive that by multiplying by 2 it becomes integrable. r l he 
other equation is already integrable, and by integrating we find 

z 2 + y * = a, xy=b, 
whence we conclude that 

z 2 + y = Pxy. 

We shall conclude what we have to say upon equations of partial differ 
ences of the first order, by the solution of this problem. 

Given an equation which contains an arbitrary function of one or more 
variables, tojind the equation of partial differences "which produced it. 
Suppose we have 

z= F(x* + y 2 ). 
Make 

x 2 + y 2 = u .......... (0 

and the equation becomes . 

z = Fu. 
49 



Hi INTRODUCTION. 

The differential of F u must be of the form f u . d u. Conse 
quently 

d z = d u. <p u 

If we take the differential of z relatively to x only, that is to say, in 
regarding y as constant, we ought to take also d u on the same 
hypothesis. Consequently, dividing the preceding equation by d x, 
we get 

d z\ /d U 



Again, considering x as constant and y as variable, we shall similarly 
find 

(} = / d \ 9 u 
Vdy/ \dy/ 

But the values of these coefficients are found from the equation (f) f 
which gives 

/d u\ /d u\ 

I j J = 2 x , ( -j ) = 2 y . 

\d x/ \d y/ * 

Hence our equations become 
fd z> 



(dz\ _ /dz\ rt 

dx) =2x ? u,( a7 )=2y ? u; 

and eliminating <p u from these, we get the equation required ; viz. 



d z\ /d 

= x 



As another example, take this equation 

z 8 + 2 ax = F (x y). 
Making 

x y = u , 

It becomes 

z s + 2ax=Fu 
and differ ntiating, we get 

d(z s -f-2ax) = du?u. 
Then taking the differential relatively to x, we have 



and similarly, with regard to y, we get 

/d z\ /d u 

82 



ANALYTICAL GEOMETRY. "liii 

But since 

x y = u 

u 



. . 


which, being substituted in the above equation, gives us 



and eliminating <p u from these, we have the equation required ; viz. 



We now come to 



EQUATIONS OF PARTIAL DIFFERENCES OF THE SECOND ORDER. 



Aii equation of Partial Differences of the second order in which z is a 
function of two variables x, y ought always to contain one or more of the 
differential coefficients 



independently of the differential coefficients which enter equations of the 
first order. 

We shall merely integrate the simplest equations of this kind, and shall 
begin with this, viz. 



Multiplying by d x and integrating relatively to x we add to the inte 
gral an arbitrary function of y ; and we shall thus get 



/dz\ 

(die) 



Again multiplying by d x and integrating, the integral will be com 
pleted when we add another arbitrary function of y, viz. -^ y. We thus 
obtain 

z = x p y -f ^ y. 

Now let us integrate the equation. 



P 
d3 



liv INTRODUCTION. 

in which P is any function of x, y. Operating as before we first obtain 

(d z\ 
diJ =/Pdx + ?y; 

and the second integration gives us 

z = //Pdx -f- 9 y] dx 
In the same manner we integrate 

P 

- * 



^dy 
and find 

The equation 



II 

must be integrated first relatively to one of the variables, and then rela 
tively to the other, which will give 



y + /Pdx}dy . 

In general, similarly may be treated the several equations 
_ p 



in which P, Q, R, &c. are functions of x, y, which gives place to a series 
i integrations, introducing for each of them an arbitrary function. 
One of the next easiest equations to integrate is this 



(af) = 

which P and Q will always denote two functions of x and y. 
Make 



d 

= U 



and the proposed will transform to 



To integrate this, we consider x constant, and then it contains only 
two variables y and u, and it will be of the same form as the equation 

dy + Pydx = Qdx 
whose integral (see Vol. 1. p. 109) is 



y = e -/*ax {/Qe/"dx + CJ. 
Hence our equation gives 



u =e- 



ANALYTICAL GEOMETRY. 

But 

U = 



Hence by integration we get 

z =f{ e- pd y (/Qe PJ ydy) + p x } dy + -^x. 
By the same method we may integrate 

-p. /d z\ ~ d 2 z -p, /d z\ ~ 

+ p (dx) = Q - a^u + p Civ) = Q> 



K! y> 

in which P, Q represent functions of x, and because of the divisor d x d y, 
we perceive that the value of z will not contain arbitrary functions of the 
same variable. 



THE DETERMINATION OF THE ARBITRARY FUNCTIONS WHICH ENTEll 

THE INTEGRALS OF EQUATIONS OF PARTIAL, DIFFERENCES O.Y 

THE FIRST ORDEK. 



The arbitrary functions which complete the integrals of equations of 
partial differences, ought to be given by the conditions arising from the 
nature of the problems from which originated these equations ; problems 
generally belonging to the physical branches of the Mathematics. 

But in order to keep in view the subject we are discussing, we shall 
limit ourselves to considerations purely analytical, and we shall first seek 
what are the conditions contained in the equation 

/d z\ _ 
Vd x/ 

Since z is a function of x, y, this equation may be ;,msidered as that of 
a surface. This surface, from the nature of its equation, has the followino- 

property, that f-r 1 must always be constant. Hence it follows that 

every section of this surface made by a plane parallel to that of x, y is a 
straight line. In fact, whatever may be the nature of this section, if we 
divide it into an infinity of parts, these, to a small extent, may be con 
sidered straight lines, and will represent the elements of the section, or.e 
of these elements making with a parallel to the axis of abscissae, an angle 

/dz\ 

whose tangent is (7-). Since this angle is constant, it follows that all 



the angles formed in like manner by the elements of the curve, with par- 

4 4 



Ivi INTRODUCTION. 

allels to the axis of abscissae will be equal. Which proves that the sec 
tion in question is a straight line. 

We might arrive at the same result by considering the integral of the 
equation 



= a 
^u x/ 

which we know to be 

z = a x + p y, 

since for all the points .of the surface which in the cutting plane, the or- 
dinate is equal to a constant c. Replacing therefore p y by p c, and 
making p c = C, the above equation becomes 

z = a x + C ; 

this equation being that of a straight line, shows that the section is a 
straight line. 

The same holding good relatively to other cutting planes which may be 
drawn parallel to that of x, z, we conclude that all these planes will cut the 
surface in straight lines, which will be parallel, since they will each form 
with a parallel to the axis of x, an angle whose tangent is a. 

If, however, we make x = 0, the equation z = a x + p y reduces to 
z =: Py an d will be that of a curve traced upon the plane of y, z; this 
curve containing all the points of the surface whose coordinates are x = 0, 
will meet the plane in a point whose coordinate is x =0; and since we 
have also y = c, the third coordinate by means of the equation 

z = ax + C 

will be 

z = C. 

What has been said of this one plane, applies equally to all others 
which are parallel to it, and it thence results that through all the points 
of the curve whose equation is z = p y, and which is traced in the plane 
of y, z, will pass straight lines parallel to the axis of x. This is ex 
pressed by the equations 

d z> 



/ z\ 

( -r- ) = 
\d x/ 



and 

Z = ax + py; 

and since this condition is always fulfilled, whatever may be the figure of 
the curve whose equation is z <p y, we see that this curve .is arbi 
trary. 

From what precedes, it follows that the curve whose equation is z = py 



ANALYTICAL GEOMETRY. 



Ivii 



may be composed of arcs of different curves, which unite at their extre 
mities, as in this diagram 




or which have a break off in their course, as in this figure. 

,N 




In the first case the curve is discontinuous^ and in the second it is dis 
contiguous. We may remark that in this last case, two different ordinates 
P M, P N corresponding to the same abscissa A P; finally, it is possible, 
that without being discontiguous, the curve may be composed of an in 
finite series of arcs indefinitely small, which belong each of them to 
different curves ; in this case, the curve is irregular, as will be, for 
instance, the flourishes of the pen made at random ; but in whatever way 
it is formed, the curve whose equation is z = <p y, it will suffice, to con 
struct the surface, to make a straight line move parallelly with this condi 
tion, that its general point shall trace out the curve whose equation is 



z = 



and vhich is traced at random upon the plane of y, z. 
If instead of the equation 

/d z\ 

(di) = a 
we had 

/d z\ Y 

Id x) ~ X > 

in which X was a function of x, then in drawing a plane parallel to the 
plane (x, z), the surface will be cut by it no longer in a straight line, as 
in the preceding case. In fact, for every point taken in this section, the 
tangent of the angle formed by the element produced of the section, with 
a parallel to the axis of x, will be equal to a function X of the abscissa x 
of this point; and since the abscissa x is different for overy point :t foJ- 



Iviii .. INTRODUCTION. 

lows that this angle will be different at each point of the section, which 
section, therefore, is no longer, as before, a straight line. The surface 
will be constructed, as before, by moving the section parallelly, so that its 
point may ride continually in the curve whose equation is z = <p y. 

Suppose now that in the preceding equation, instead of X we have a 
function, P of x, and of y. The equation 

(T- Z ) = p 

VI x/ 

containing three variables will belong still to a curve surface. If we cut 

O . 

this surface by a plane parallel to that of x, z, we shall have a section in 
which y will be constant ; and since in all its points (j^) wi ^ be ec l ual 

to a function of the variable x, this section must be a curve, as in the pre 
ceding case. The equation 

(i- z ) = P 

VI x/ 
being integrated, we shall have for that of the surface 

z =/Pdx + py; 

if in this equation we give successively to y the increasing values y , y", 
y ", &c. and make P, P , P" , &c. what the function P becomes in these 
cases, we shall have the equations 

z = /P dx + y , z =/P"dx + py" 1 

z = /P "dx + py" , z = /P""dx + py"" &c. / 
and we see that these equations will belong to curves of the same nature, 
but different in form, since the values of the constant y will not be the 
same. These curves are nothing else than the sections of the surface 
made by planes parallel to the plane (x, z) ; and in meeting the plane 
(y, z) they will form a curve whose equation will be obtained by equating 
to zero, the value of x in that of the surface. Call the value of/Pdx, 
in this case, Y, and we shall have 

z = Y + py; 

and we perceive that by reason of p y, the curve determined by this equa.- 
tion must be arbitrary. Thus, having traced at pleasure a curve, Q R S, 
upon the plane (y, z), if we represent by R L the section whose equation 

Q 

L 




is z = f P d x -f f> y , we shall move this section, always keeping the ex- 



ANALYTICAL GEOMETRY. lix 

tremity R applied to the curve Q R S ; but so that this section as it 
moves, may assume the successive forms determined by the above group 
of equations, and we shall thus construct the surface to which will belong 
the equation 



Era = R 

Finally let us consider the general equation 



whose integral is U = <p V. Since U = a, V = b, each of these equa 
tions subsisting between three coordinates, we may regard them as be 
longing to two surfaces ; and since the coordinates are common, they 
ought to belong to the curve of intersection of the two surfaces. This 
being shown, a and b being arbitrary constants, if in U = a, we give to 
X and y the values x , y we shall obtain for z, a function of x , of y and 
of a, which will determine a point of the surface whose equation is U = a. 
This point, which is any whatever, will vary in position if we give succes 
sively different values to the arbitrary constant a, which amounts to say 
ing that by making a vary, we shall pass the surface whose equation is 
U = a, through a new system of points. This applies equally to V = b, 
and we conclude that the curve of intersection of the two surfaces will 
change continually in position, and consequently will describe a curved 
surface in which a, b may be considered as two coordinates ; and since 
the relation a = <p b which connects these two coordinates, is arbitrary 
we perceive that the determination of the function <p amounts to making 
a surface pass through a curve traced arbitrarily. 

To show how this sort of problems may conduct to analytical condi 
tions, let us examine what is the surface whose equation is 
d z\ /d 



= x 



We have seen that this equation being integrated gives 

z = p(x 2 + y 1 ). 
Reciprocally we hence derive 

x* + y j = 0>z. 

If we cut the surface by a plane parallel to the plane (x, y) the equation 
of the section will be 

x 2 + y* = <& c; 
and representing by a * the constant 4> c, we shall have 

x 1 + y 2 = a 2 . 
This equation belongs to the circle. Consequently the surface will 



U 



INTRODUCTION. 



have this property, viz. that every section made by a plane parallel to the 
plane (x, y) will be a circle. 

This property is also indicated by the equation 
d 



for this equation gives 

dy 
x = y -J-&. 

J d x 

This equation shows us that the subnormal ought to be always equal to 
the abscissa which is the property of the circle. 

The equation z = <p (x 2 + y*) showing merely that all the sections 
parallel to the plane (x, y) are circles, it follows thence that the law ac 
cording to which the radii of these sections ought to increase, is not 
comprised in this equation, and that consequently, every surface of revo 
lution will satisfy the problem ; for we know that in this sort of surfaces, 
the sections parallel to the plane (x, y) are always circles, and it is need 
less to say that the generatrix which, during a revolution, describes the 
surface, may be a curve discontinued, discontiguous, regular or irregular. 

Let us therefore investigate the surface for which this generatrix will 
be a parabola A N, and suppose that, in this hypothesis, the surface is 
cut by a plane A B, which shall pass through the axis of z , the trace of 



B 




Q 



this plane upon the plane (x, y) will be a straight line A L, which, being 
drawn through the origin, will have the equation y = a x ; if we repre 
sent by t the hypothenuse of the right angled triangle A P Q, constructed 
upon the plane (x, y) we shall have 

t j = x 2 + y a i 

but t being the abscissa of the parabola A M, of which Q M = 2 i the 
ordinate, we have, by the nature of the curve, 

t* = bz. 
Putting for t 2 its value x * + y ! , we get 



Z = (y 



orz 



= ^x( + n ); 



ANALYTICAL GEOMETRY. ixi 

- and making 

i (a + 1) = m, 

we shall obtain 

z = mx 2 ; 

so that the condition prescribed in the hypothesis, where the generatrix 
is a parabola, is that we ought to have 

z = m x *, when y = a x. 

Let us now investigate, by means of these conditions, the arbitrary 
function which enters the equation z = <f> (x ! + y *). For that pur 
pose, we shall represent by U the quantity x * + y 3 which is effected by 
the symbol <p, and the equation then becomes 

z = f> U; 
and we shall have the three equations 

x * + y * = U, yrrax, z = m x *. 

By means of the two first we eliminate y and obtain the value of x * 
which being put into the third, will give 

Z = m . ^ - - r 
1 + a* > 

an equation which reduces to 

7, TT 

b" U> 

the value of z being substituted in the equation z = <f> U, will change 
it to 



and putting the value of U in this equation, we shall find that 



and we see that the function is determined. Substituting this value of 
<p (x z + y 9 ) in the equation z <f> (x 2 + y 2 ), we get 

Z= b (x * + y2) 

for the integral sought, an equation which has the property required, 
since the hypothesis of y = ax gives 

z = m x *. 

This process is general ; for, supposing the conditions which determine 
the arbitrary constant to be that the integral gives F (x, y, z} = 0, when 
we have f (x, y, z) = 0, we shall obtain a third equation by equating to 



ixii 



INTRODUCTION. 



CJ the quantity which follows p, and then by eliminating, successively, 
two of the variables x, y, z, we shall obtain each of these variables in a 
function of U ; putting these values in the integral, we shall get an equa 
tion whose first member is <p U, and whose second member is a compound 
expression in terms of U ; restoring the value of U in terms of the vari- 
bles, the arbitrary function will be determined. 



THE ARBITRARY FUNCTIONS WHICH ENTER THE INTEGRALS OF THE 
EQUATIONS OF PARTIAL DIFFERENCES OF THE SECOND ORDER. 

Equations of partial differences of the second order conduct to integrals 
which contain two arbitrary functions ; the determination of these func 
tions amounts to making the surface pass through two curves which may 
be discontinuous or discontiguous. For example, take the equation 



whose integral has been found to be 



Let A x, A y, A z, be the axis of coordinates; if we draw a plane 




K L parallel to the plane (x, z), the section of the surface by this plane 
will be a straight line ; since, for all the points of this section, y being 
equal to A p, if we represent A p by a constant c, the quantities <p y, ^ y 
will become <p c, -^ c, and, consequently, may be replaced by two con 
stants, a, b, so that the equation 

z = x py -f 4y 



ANALYTICAL GEOMETRY. Ixiii 

will become 

z = a x -f- b, 
and this is the equation to the section made by the plane K L. 

To find the point where this section meets the plane (y, z) make 
x = 0, and the equation above gives z = -4/ y, which indicates a curve 
a m b, traced upon the plane (y, z). It will be easy to show that the 
section meets the curve a m b in a point m ; and since this section is a 
straight line, it is only requisite, to find the position of it, to find a second 
point. For that purpose, observe that when x = 0, the first equation 
reduces to 

z = ^y, 

whilst, when x = 1, the same equation reduces to 

z = 9 y + 4- y- 

Making, as above, y = Ap = c, these two values of z will become 

z = b, z = a -f b, 

and determining two points m and r, taken upon the same section, in r 
we know to be in a straight line. To construct these points we thus pro 
ceed : we shall arbitrarily trace upon the plane (y, z) the curve a m b, 
and through the point p, where the cutting plane K L meets the axis of 
y, raise the perpendicular pm = b, which will be an ordinate to the 
curve ; we shall then take at the intersection H L of the cutting plane, 
and the plane (x, y), the part p p equal to unity, and through the point 
p , we shall draw a plane parallel to the plane (y, z), and in this plane 
construct the curve a m b , after the modulus of the curve a m b, and so 
as to be similarly disposed ; then the ordinate m p will be equal to m p ; 
and if we produce m p by m r, which will represent a, we shall deter 
mine the point r of the section. 

If, by a second process, we then produce all the ordinates of the curve 
a m b , we shall construct a new curve a r b , which will be such, that 
drawing through this curve and through a m b, a plane parallel to the 
plane (x, z), the two points where the curves meet, will belong to the 
same section of the surface. 

From what precedes, it follows that the surface may be constructed, by 
moving the straight line m r so as continually to touch the two curves, 
a m b, a m b . 

This example suffices to show that the determination of the arbitrary 
functions which complete the integrals of equations of partial differences 
of the second order, is the same as making the surface pass through two 
curves, which, as well as the functions themselves, may be discontinuous, 
discontiguous, regular or irreguiar. 



Ixir INTRODUCTION 



CALCULUS OF VARIATIONS. 



If we have given a function Z F, (x, y, y , y"), wherein y , y" mean 



y itself being a function of x, it may be required to make L have certain 
properties, (such as that of being a maximum, for instance) whether by 
assigning to x, y numerical values, or by establishing relations between 
these variables, and connecting them by equations. When the equation 
y = p x is given, we may then deduce y, y 7 , y" . . . in terms of x and sub 
stituting, -we have the form 

Z = f x. 

By the known rules of the differential calculus, we may assign the values 
ofx, when we make of x a maximum or minimum. Thus we determine what 
are the points of a given curve, for which the proposed function Z, is 
greater or less than for every other point of the same curve. 

But if the equation y = <p x is not given, then taking successively for 
<f> x different forms, the function Z = f x will, at the same time, assume 
different functions of x. It may be proposed to assign to f x such a 
form as shall make Z greater or less than every other form of p ^for the 
same numerical value ofx whatever it may be in other respects. This latter 
species of problem belongs to the calculus of variations. This theory 
relates not to maxima and minima only; but we shall confine our 
selves to these considerations, because it will suffice to make known all 
the rules of the calculus. We must always bear in mind, that the varia 
bles x, y are not independent, but that the equation y = px is unknown, 
and that we only suppose it given to facilitate the resolution of the prob 
lem. We must consider x as any quantity whatever which remains the same 
for all the differential forms of <p x ; the forms of <p, p , <p" . . . . are therefore 
variable, whilst x is constant. 

In Z = F (x, y, y , y". . .) put y + k for y, y + k , for y . . . , k being 
an arbitrary function of x, and k , k./ . . . the quantities 

dl^ dMc 
dV dx*" 
But, Z will become 

Z, = F (X, y -f k, y + k , y" + k," . . .) 



ANALYTICAL GEOMETRY. Ixv 

Taylor s theorem holds good whether the quantities x, y, k be depen 
dent or independent. Hence we have 



so that we may consider x, y, y , y" . . . as so many independent variables. 
The nature of the question requires that the equation y = <p x should 
he determined, so that for the same value of x, we may have always 
Z 7 > Z, or Z / < Z : reasoning as in the ordinary maxima and minima, 
we perceive that the terms of the first order must equal zero, or that we 
have 



Since k is arbitrary for every value of x, and it is not necessary that its 
value or its form should remain the same, when x varies or is constant, 
k , k" . . . is as well arbitrary as k. For we may suppose for any value 
x = X that k = a + b (x X) + | c (x X) * + &c., X, a, b, c . . . 
being taken at pleasure ; and since this equation, and its differentials, 
ought to hold good, whatever is x, they ought also to subsist when 
x = X, which gives k = a, k = b, k" = c, &c. Hence the equation 
Z, = Z + . . . cannot be satisfied when a, b, c . . . are considered inde 
pendent, unless (see 6, vol. I.) 

/d Zx /d Zx /d Z N / d Z 

3p = (ay) == (37 ) = v- (dyrn 

n being the highest order of y in Z. These different equations subsist 
simultaneously, whatever may be the value of x ; and if so, there ought 
to be a maximum or minimum ; and the relation which then subsists be 
tween x, y will be the equation sought, viz. y = <p x, which will have the 
property of making Z greater or less than every other relation between 
x and y can make it. We can distinguish the maximum from the mini 
mum from the signs of the terms of the second order, as in vol. I. 
p. (31.) 

But if all these equations give different relations between x, y, the 
problem will be impossible in the state of generality which we have 
ascribed to it ; and if it happen that some only of these equations subsist 
mutually, then the function Z will have maxima and minima, relative to 
some of the quantities y, y , y" . . . without their being common to them 
all. The equations which thus subsist, will give the relative maxima and 
minima. And if we wish to make X a maximum or minimum only relatively 



ixvi INTRODUCTION. 

to one of the quantities y, y , y" . . . , since then we have only one equa 
tion to satisfy, the problem will be always possible. 

From the preceding considerations it follows, that first, the quantities 
X, y depend upon one another, and that, nevertheless, we ought to make 
them vary, as if they were independent, for this is but an artifice to get 
the more readily at the result. 

Secondly, that these variations are not indefinitely small ; and if we em 
ploy the differential calculus to obtain them, it is only an expeditious 
means of getting the second term o the developement, the only one 
which is here necessary. 

Let us apply these general notions to some examples. 

Ex. 1. Take, upon the axis of x of a curve, two abscissas m, n; and 
draw indefinite parallels to the axis of y. Let y = <p x be the equation 
of this curve: if through any point whatever, we draw a tangent, it will 
cut the parallels* in points whose ordinates are 

1 = y + y (m x), h = y + y (n x) . 

If the form of 9 is given, every thing else is known; but if it is not 
given, it may be asked, what is the curve which has the property of 
having for each point of tangency, the product of these two ordinates less 
than for every other curve. 

Here we have 1 X h ; or 

Z = { y X (m x) y } + { y + (n - x) y J . 

From the enunciation of the problem, the curves which pass through the 
same point (x, y) have tangents taking different directions, and that which 
is required, ought to have a tangent, such that the condition Z = maximum 
is fulfilled. We may consider x and y constant ; whence 

/d_Z\ . 2y _ 2 x in n 1 1 

V d yv " y ~ (x m) (x n) ~ x mx n* 

Then integrating we get 

y 2 = C(x m) (x n). 

The curve is an ellipse or a hyperbola, according as C is positive or 
negative ; the vertexes are given by x = m, x =s n ; in the first case, the 
product h X 1 or Z is a maximum, because y" is negative; in the second, 
Z is a minimum or rather a negative maximum ; this product is moreover 
constant, and 1 h = 1 C (m n) 2 , the square of the semi-axis. 

Ex. 2. What is the curve for which, in each of its points, the square of 
the subnormal added to the abscissa is a minimum ? 

We have in this case 

Z = (y y + x) 2 



ANALYTICAL GEOMETRY. Ixvii 

whence \ve get two equations subsisting mutually by making 

y y + x = 
and thence 

x 2 + y 2 = r \ 

Therefore all the circles described from the origin as a center wi" 1 alone 
satisfy the equation. 

The theory just expounded has not been greatly extended ; but it serves 
as a preliminary developement of great use for the comprehension of a 
far more interesting problem which remains to be considered. This re 
quires all the preceding reasonings to be applied to a function of the form 
/* Z: the sign y indicates the function Z to be a differential and that after 
having integrated it between prescribed limits" it is required i,o endow it 
with the preceding properties. The difficulty here to be overcome is that 
of resolving the problem without integrating. 

When a body is in motion, we may compare together either the differ 
ent points of the body in one of its positions or the plane occupied suc 
cessively by a given point. In the first case, the body is considered fixed, 
and the symbol d will relate to the change of the coordinates of its surface; 
in the second, we must express by a convenient symbol, variations alto 
gether independent of the first, which shall be denoted by 8. When we 
consider a curve immoveabie, or even variable, but taken in one of its po 
sitions, d x, d y . . . announce a comparison between its coordinates ; but 
to consider the different planes which the same point of a curve occupies, 
the curve varying in form according to any law whatever, we shall write d 
x, 5 y ... which denote the increments considered under this point of view, 
and are functions of x, y . . . In like manner, d x becoming d (x + <3 x) 
will increase by d 5 x ; d 2 x will increase by d 2 3 x, &c. 

Observe that the variations indicated by the symbol <3 are finite, and 
wholly independent of those which d represents ; the operations to which 
these symbols relate being equally independent, the order in which they 
are used must be equally a matter of indifference as to the result. So 
that we have 

<5.d x = d. 5 x 
d 2 . 8 x = 3 . d 2 x 
&c. 

/a U = * - U. 

and so on. 

It remains to establish relations between x, y, 7. . .such that/Z may 
be a maximum or a minimum letween given limits. That the calculus may 
he rendered the more symmetrical, we shall not suppose any differential 



Ixviu INTRODUCTION 

constant ; moreover we shall only introduce three variables because it will 
be easy to generalise the result. To abridge the labour of the process, 

make 

d x = x /5 d 2 x x //5 &c. 

so that 

z = F (x, x,, x /y , . . . y, y y , y lfl . . . z, z,, z,, . . .). 

Now x, y and z receiving the arbitrary and finite increments 3 x, d y, 
8 z, d x or x, becomes 

d (x + a x) = d x + a d x or x, + 6 x,. 

In the same manner, x,, increases by a x,, and so on ; so that develop 
ing Z, by Taylor s theorem, and integrating / Z becomes 



The condition of a maximum or minimum requires the integral of the 
terms of the first order to be zero between given limits whatever may be 
ii x, B y, d z as we have already seen. Take the differential of the known 
function Z considering x, x /} X// . . . y, y,, y,, ... as so many independent 
variables ; we shall have 

dZ = mdx4-ndx + p d x + . . . M d y + N d y x . . . + /A d z + v d z / . . . 
m n ... M, N .../*, v ... being the coefficients of the partial differences 
of Z relatively to x, x 7 . . . y, y, . . . z, z /} . . . considered as so many varia 
bles ; these are therefore known functions for each proposed value of Z. 
Performing this differentiation exactly in the same manner by the symbol 
3, we have 




But this known quantity, whose number of terms is limited, is precisely 
that which is under the sign /, in the terms of the first order of the de- 
velopement : so that the required condition of max. or min. is that 

/3Z = 0, 

between given limits, whatever may be the variations 5 x, 8 y, d z. Ob 
serve, that here, as before, the differential calculus is only employed as a 
means of obtaining easily the assemblage of terms to be equated to zero ; 
so that the variations are still any whatever and finite. 



ANALYTICAL GEOMETRY. 

We have said that d . 8 x may be put for d . 3 x ; thus the first line is 
equivalent to 

m, n . . . contains differentials, so that the defect of homogeneity is here 
only apparent. To integrate this, we shall see that it is necessary to 
disengage from the symbol f as often as possible, the terms which con 
tain d 3. To effect this, we integrate by parts which gives 

y n d 3 x = n . 3x yd n . 3 x 

/p.d 2 3x = p d 3x d p 3x+/d p3x 

yqd 3 5x = qd 2 3x dq.d3x-f- d 2 q.dx f d 3 q . 3 x 

&c. 

Collecting these results, we have this series, the law of which is easily 
recognised ; viz. 

/ (m d n + d * p d 3 q + d 4 r . . .)3x 

-f- (p d q + d 2 r d 3 s -f d 4 t . . .) d 3 x 
+ (q d r + . . .) d 2 3 x 
+ &c. 
The integral of (A) ory. 3 z = , becomes therefore 

i- d n +d 8 p-...)3 x + (M-d N+d 2 P-...)3 y+ (//-d v-...)8 z] =0 



C 

J 
(. 



+ (q-dr...) d 2 3x ...+ K = 

K being the arbitrary constant. The equation has been split into two, 
because the terms which remain under the sign y cannot be integrated, at 
least whilst 3 x, 3 y, 8 z are arbitrary. In the same manner, if the nature 
of the question does not establish some relation between 3 x, 3 y, 3 z, the 
independence of these variations requires also that equation (B) shall again 
make three others ; viz. 

0=m dn + d 2 p d j q -f- d 4 r 1 
= M dN+d 2 P d 3 Q+d 4 R .... S- . . (D) 

Consequently, to find the relations between x, y, z, which make y Z a 
maximum, we must take the differential of the given function Z by con 
sidering x, y, z, d x, d y, d z, d z x, . . . as so many independent vari 
ables, and use the letter 3 to signify their increase; this is what is termed 
taking the variation of Z. Comparing the result with the equation (A), 
we shall observe the values of m, M, /i, n, N ... in terms of x, y, z, and 

e3 



LXK INTRODUCTION. 

their differences expressed by d. We must then substitute these in the 
equations (C), (D) ; the first refers to the limits between which the 
maximum should subsist ; the equations (D) constitute the relations re 
quired; they are the differentials of x, y, z, and, excepting a case of 
absurdity, may form distinct conditions, since they will determine nume 
rical values for the variables. If the question proposed relate to Geo 
metry, these equations are those of a curve or of a surface, to which 
belongs the required property. 

As the integration is effected and should be taken between given limits, 
the terms which remain and compose the equation (C) belong to these 
limits : it is become of the form K + L = 0, L being a function of 
x, y, z, 8 x, 8 y, d z . . . Mark with one and two accents the numerical 
values of these variables at the first and second limit. Then, since the 
integral is to be taken between these limits, we must mark the different 
terms of L which compose the equation C, first with one, and then with 
two accents ; take the first result from the second and equate the differ 
ence to zero ; so that the equation 

L /x - L, = 

contains no variables, because x, d x . . . will have taken the values 
x /} 3 x / . . . x //5 o x 7/ . . . assigned by the limits of the integration. We 
must remember that these accents merely belong to the limits of the 
integral. 

There are to be considered four separate cases. 

1. If the limits are given andjixed, that is to say, if the extreme values 
of x, y, z are constant, since a x,, d 8 x, . . . d x,,, d 8 x /x , &c. are zero, all 
the terms of L, and L,, are zero, and the equation (C) is satisfied. Thus 
we determine the constants which integration introduces into the equations 
(D), by the conditions conferred by the limits. 

2. If the limits are arbitrary and independent, then each of the coeffi 
cients a x, , 3 x /y . . . in the equation (C) is zero in particular. 

3. If there exist equations of condition, (which signifies geometrically 
that the curve required is terminated at points which are not fixed, but 
which are situated upon two given curves or surfaces,) for the limits, that 
is to say, if the nature of the question connects together by equations, 
some of the quantities x,, y /5 z /} x //} y,,, z /7 we use the differentials of these 
equations to obtain more variations d x,, 3 y,, 3 z,, d x ;/ , &c. in functions 
of the others; substituting in L /7 L, = 0, these variations will be re- 
duced to the least number possible : the last being absolutely independent, 
the equation will split again into many others by equating separately their 
coefficients to zero. 



ANALYTICAL GEOMETRY. Ixxi 

Instead of this process, we may adopt the following one, which is more 
elegant. Let 

u = 0, v = 0, &c. 

be the given equations of condition ; we shall multiply their variations 
t u, 3 v ... by the indeterminates X, X . . . This will give Xdu + X Sv + ... 
a known function of d x /} 6 x //5 d y, . . . Adding this sum to L x/ L,, we 
shall get 

L,, L, + X d u + X d v + . . . = . . . . (E). 
Consider all the variations 8 x /} d x //} ... as independent, and equate 
their coefficients separately to zero. Then we shall eliminate the inde 
terminates X, X . . . from these equations. By this process, we shall arrive 
at the same result as by the former one ; for we have only made legiti 
mate operations, and we shall obtain the same number of final equations. 

It must be observed, that we are not to conclude from u = 0, v = 0, 
that at the limits we have d u = 0, d v = ; these conditions are inde 
pendent, and may easily not coexist. In the contrary case, we must 
consider d u = 0, d v = 0, as new conditions, and besides X d u, we 
must also take X d d u . . . 

4. Nothino- need be said as to the case where one of the limits is fixed 

O 

and the other subject to certain conditions, or even altogether arbitrary, 
because it is included in the three preceding ones. 

It may happen also that the nature of the question subjects the varia 
tions o X, d y, d z, to certain conditions, given by the equations 

i - 0, 6 = 0, 

and independently of limits; thus, for example, when the required curve 
is to be traced upon a given curve surface. Then the equation (B) will 
not split into three equations, and the equations (D) will not subsist. We. 
must first reduce, as follows, the variations to the smallest number possi 
ble in the formula (B), by means of the equations of condition, and equate 
to zero the coefficients of the variations that remain ; or, which is tanta 
mount, add to (B) the terms X5? + X 60 + ...; then split this equation 
into others by considering d x, 6 y, 3 z as independent ; and finally elimi 
nate X, X ... 

It must be observed, that, in particular cases, it is often preferable to 
make, upon the given function Z, all the operations which have produced 
the equations (B), (C) instead of comparing each particular case with the 
general formulae above given. 

Such are the general principles of the calculus of variations: let us 
illustrate it with examples. 



Ixxii 



INTRODUCTION. 



Ex. 1. What is the curve C M K of which the length M K, comprised 
between the given radii-vectors A M, A K is the least possible. 




We have, (vol. I, p. Q00)> if r be the radius-vector, 
s = /(r ! d<?* + d 2 ) = Z 

it is required to find the relation r = <p 6, which j-end&rs Z a minimum 
the variation is 

7 _ r d <? 2 . a .f r 2 d 4. ad d + dr . od r 
V (r * d 6 + d r ) 

Comparing with equation (A) ; where we suppose x = r, y = 6 t we 
have 

r d 6 z d r r * d d 

m = j , n = . , M = , N = , 

as d s d s 

the equations (D) are 

r d 6* 



ds 



T 

d s 



_ 

C 



Eliminating d 0, and then d s, from these equations, and d s 2 = r * d P; 
4- d r 2 , we perceive that they subsist mutually or agree; so that it is 
sufficient to integrate one of them. But the perpendicular A I let fall 
from the origin A upon any tangent whatever. T M is 
A J = A M + sin. A M T = r sin. /?, 
which is equivalent, as we easily find, to 

r tan. 



which gives 



V (1 + tan. 2 /3) 



d 6 



V (r * d 6* + d r ) ~dl~ 

and since this perpendicular is here constant, the required line is a 
straight line. The limits M and K being indeterminate, the equations 
(C) are unnecessary. 

Ex. 2. To Jind the shortest line between two given points, or two given 



curves. 



ANALYTICAL GEOMETRY. Ixxiii 

The length" s of the line is 

/Z =fV (dx 2 + dy* + dz 2 ). 
It is required to make this quantity a minimum ; we have 

, ,7 dx, , d y . , dz.. 

o L =. -, a d x + -r-^ d y + -5 <5 d z, 
d s d s d s 

ind comparing with the formula (A), we find 

rti\T d x XT d y dz 

m = 0,M = 0,/A = 0,n= , , N = ~^-- , v sr -^ : 

as as d s 

the other coefficients P, p, * . . . are zero. The equations (D) become* 
therefore, in this case, 

whence, by integrating 

Squaring and adding, we get 

a+ b 2 + c 2 = 1, 

a condition that the constants a, b, c must fulfil in order that these equa 
tions may simultaneously subsist. By division, we find 

d y _ b d z _ c_ 
d x ~~ a oTx ~ a* 
whence 

b x = a y + a , c x = a z + b ; 

the projections of the line required are therefore straight lines the line is 
therefore itself a straight line. 

To find the position of it, we must know the five constants a, b, c, 
a , b . If it be required to find the shortest distance between two given 
fixed points (x , y,, z,), (X A , y //} zj, it is evident that a, x, a x /7 , ay,... are 
zero, and that the equation (C) then holds good. Subjecting our two 
equations to the condition of being satisfied when we substitute therein 
x / x /, y/ f r x / y/ z, we shall obtain four equations, which, with 
a 2 + b a + c 2 = 1, determine the five necessary constants. 

Suppose that the second limit is a fixed point (x //? y //} z /7 ), in the plane 
(x, y), and the first a curve passing through the point (x /5 y y z ; ), and also 
situated in this plane ; the equation 

b x =r a y + a 
then suffices. Let y, = f x, be the equation of the curve ; hence 

a y/ = A3 X/; 

the equation (C) becomes 



Ixxiv INTRODUCTION. 

and since the second limit is fixed it is sufficient to combine together the 

equations 

dy, = ASx, 
dx,3x, + dy / 5y / =r 0. 
Eliminating d y, we get 

dx, + Ady, = 0. 

We might also have multiplied the equation of condition 

S y, A S x, = 

by the indeterminate X, and have added the result to L,, which would 
have given 

(a-f) 8x + On) Sy + ^y<- xASx < = 0> 

whence 

_ x A = 0, (^ ) + x = 0. 



d s 

Eliminating X we get 

dx, + Ad y/ = 0. 

But then the point (x /} y,) is upon the straight line passing through the 
points ( X/ , y/ , Z/ ), {x //$ y// , Z// ), and we have also 

b d x, = ad y /} 
whence 

a = b A 
and 

ly = - -1 = - ; 

dx A a 

which shows the straight line is a normal to the curve of condition. The 
constant a is determined by the consideration of the second limit which is 
given and fixed. 

It would be easy to apply the preceding reasoning to three dimensions, 
and we should arrive at similar conclusions; we may, therefore, infer 
generally that the shortest distance between two curves is the straight 
line which is a normal to them. 

If the shortest line required were to be traced upon a curve surface 
whose equation is u - 0, then the equation (B) would not decompose into 
three others. We must add to it the term X d u ; then regarding 6 x, 5 y, 
fi z as independent, we shall find the relations 



ANALYTICAL GEOMETRY. 



Ixxv 



* 



From these eliminating A, we have the two equations 
d u\ dxv du\ , /d 



/d u\ , /xv /u 

(dz) d .(dl)= (<Tx 

d z\ /du\t 



\vhich are those of the curve required. 

Take for example, the least distance measured upon the surface of a 




1C 



sphere, whose center is at the origin of coordinates : hence 
u = x, 2 + y + z* r 2 = 



=2x, = 

^d y/ 

Our equations give, making d s constant, 



whence 

y d z x = x d * y. 

Integrating we have 
zdx xdz = ads, zdy ydz = bds, ydx xdy = cds. 

Multiplying the first of these equations by y, the second by x, the 
third by z, and adding them, we get 

ay = bx + cz 

the equation of a plane passing through the origin of coordinates. Hencf 
the curve required is a great circle which passes through the points A 
C , or which is normal to the two curves A B and C D which are limits 
and are given upon the spherical surface. 

When a body moves in a fluid it encounters a resistance which ceteris 



Ixxvi INTRODUCTION. 

paribus depends on its form (see vol. I.) : if the body be one of revolu 
tion and moves in the direction of its axis, we can show by mechanics 
that the resistance is the least possible when the equation of the gener 
ating curve fulfils the condition 

/y d d y 3 
* , . J , = minimum, 
d x* + d y 2 

or 



1+ y 2 
Let us determine the generating curve of the solid of least resistance 

(see Principia, vol. II.). 

Taking the variation of the above expression, we get 



. y /3dx 



(i+y 2 ) s 
the second equation (D) is 

M dN = 0; 
and it follows from what we have done relatively to Z, that 

= y dN+ Nd/, 



& c 








because 

M = d N. 
Thus integrating, we have 



3 



.4.XXLZ- - N v ~ y_y_^__o . 

14. y~ ~ L y (1 + y 2 )* 
Therefore 

a (I + y /2 ) 2 = 2yy 3 . 

Observe that the first of the equations (D) or m d n = 0, would 
have given the same result n = a ; so that these two equations conduct 
to the same result. We have 

a (1 +_/^)J 

y- --^y> 






substituting for y its value, this integral may easily be obtained ; it remains 
to eliminate f from these values of x and y, and we shall obtain the 
equation of the required curve, containing two constants which we shall 
determine from the given conditions. 



ANALYTICAL GEOMETRY. Ixxvii 

Ex. 3. WJiat is the curve ABM in which the area B O D M comprised 




between the arc B M the radii of curvature B O, D M and the arc O D 
of the evolute, is a minimum ? 
The element of the arc A M is 



dsrrdxvM-fy ; 
the radius of curvature M D is 



and their product is the element of the proposed area, or 



^ 
y" d x d y 

It is required to find the equation y = f x, which makes f Z, a mini 
mum. 

Take the variation d N, and consider only the second of the equations 
(D), which is sufficient for our object, and we get 

M = 0, N d P = 4 a, 

XT dx* + dy 2 1 + y * 

N = -, , , J . 4 d y = ,/ 4 y , 
d x d * v J v" J 



P _ 



y /2 dx 



But 



V 



y"d x 






= 4 a d y + d P d y -f. P d y" d x, 

putting 4 a + P for N. Moreover y" d x = d y , changes the last 
terms into 

(y" d P -f P d y") d x = d (P y").. d x = d 



Ixxviii INTRODUCTION. 

Integrating, therefore, 

/I _I_ i. 2\ 2 

= a y f + b, 



2y 



_dy d 2(ay + b)dy 

A U A /I I / !!\ 2 



~2(ay + b) ~dx (1 + y") 

finally, 



On the other side we have 

y =yy d x = y x /x d y 
or 

y = y x c y f^^* d y /b d y tan.- 1 /; 

this last term integrates by parts, and we have 

y i= y 7 x c y (by a) tan.- y + f. 
Eliminating the tangent from these values of x and y, we get 

by = a( x -~c) + (b f ~ y a) / + bf, 

(by a) d x b d y a d x 

V(by-ax+g)=i -gj- -,d.= V(by l ax + g) ; 

finally, 

s = 2 V (b y a x -f g) + h. 

This equation shows that the curve required is a cycloid, whose four 
constants will be determined from the same number of conditions. 

Ex. 4. What is the curve of a given length s, between two fixed points, 
for which f y d s is a maximum ? 

We easily find 

. /d x\ , c d y 

(V + ^) ( -i ) = c , whence d x = , c , - ; Hf-t - ^ , 
u \d s/ V (y + X) * c 2 ] 

and it will be found that the curve required is a catenary. 

* is the vertical ordinate of the center of gravity of an arc 



whose length is s, we see that the center of gravity of any arc whatever of 
the catenary is lower than that of any other curve terminated by the 
same points. 

Ex. 5. Reasoning in the same way for f y * d x = minimum, and 
J y d x = const, we find y * + X y = c, or rather y = c. We have 

here a straight line parallel to x. Since ^ , is the vertical ordinate 

2/y dx 

of the center of gravity of every plane area, that of a rectangle, whose 
side is horizontal, is the lowest possible ; so that every mass of water 



ANALYTICAL GEOMETRY. Ixxix 

whose upper surface is horizontal, has its center of gravity the lowest 
possible. 



FINITE DIFFERENCES. 



If we have given a series a, b, c, d, . . . take each term of it from that 
which immediately follows it, and we shall form ihejirst differences, viz. 

a = b a, b = c b, c = d c, &c. 

In the same manner we find that this series a , b , c , d . . . gives the 
second differences 

a" = b a , b" = c b , c" = d . c , &c. 
which again give the third differences 

a " = b" a", b " = c" b", c " = d" c", &c. 
These differences are indicated by A, and an exponent being given to 
it will denote the order of differences. Thus A n is a term of the series 
of nth differences. Moreover we give to each difference, the si<m which 
belongs to it ; this is , when we take it from a decreasing series. 
For example, the function 

y = x 9x + 6 

in making x successively equal to 0, 1, 2, 3, 4 ... gives a series of 
numbers of which y is the general term, and from which we get the 
following differences, 

for x = 0, 1, 2, 3, 4, 5, 6, 7 ... 

series y = 6, 2, 4, 6, 34, 86, 168, 286 . . . 

first diff. A y = 8, 2, 10, 28, 52, 82, 118 ... 

second diff. A * y = 6, 12, 18, 24, 30, 36 ... 

third diff. A 3 y = 6, 6, 6, 6, 6, ... 

We perceive that the third differences are here constant, and that the 
second difference is an arithmetic progression : we shall always arrive at 
constant differences, whenever y is a rational and integer function of x ; 
which we now demonstrate. 

In the monomial k x m make x = a, j8, y, . . . 6, *, x (these numbers 
having h for a constant difference), and we get the series 

k m , k /3 m , . . . k 6 m , k K ra , k X m . 

Since K = X h, by developing k x m k (X h) m , and designating 
DV m, A , A" ... the coefficients of the binomial, we find, that 

k (\ _ x ") = k m h x - 1 k A h * X m ~ 2 + k A" 3 h. . . 



Ixxx INTRODUCTION. 

Such is the first difference of any two terms whatever of the series 
k m , k /3 m . . . k x m , &c, 

The difference which precedes it, or k (%, m 6 m ) is deduced by 
changing X into x and x into 6 and since x = X h, we must put 
X h for X in the second member : 
k m h(X-h) "*-i-k A h 2 (X-h m ~ 2 ) ... = k m h X **-*-{ A / + m(m-l)Jkh X m ~ 2 ... 

Subtracting these differences, the two first terms will disappear, and 
we get for the second difference of an arbitrary rank 

k m (m l)h 2 x m - 2 + kB h 3 x m - 3 + ... 

In like manner, changing X into X h, in this last developement, and 
subtracting, the two first terms disappear, and we have for the third 
difference 

km (m 1) (m 2) h X"- 3 -f k B" h 4 X m -*. . . , 
and so on continually. 

Each of these differences has one term at least, in its developement, 
like the one above ; the first has m terms ; the second has m 1 terms; 
third, m 2 terms ; and so on. From the form of the first term, which 
ends by remaining alone in the mth difference, we see this is reduced to 
the constant 

1 . 2 . 3 . . . in k h m . 

If in the functions M and N we take for x two numbers which give the 
results m, n ; then M + N becomes m + n. In the same manner, let 
m , n be the results given by two other values of x ; the first difference, 
arising from M -f N, is evidently 

(m m ) -f (n n ). 

that is, the difference of the sum is the sum of the differences. The same 
may be shown of the 3d and 4th . . . differences. 

Therefore, if we make 

x = a > & 7 ... 
in 

k x m + p x m ~ 1 + . . . 

the mth difference will be the same as if these were only the first term 
k x m , for that of p x m *, q x m ~ 2 ... is nothing. Therefore the mth 
difference is constant, lichen for x ive substitute numbers in arithmetic pro 
gression, in a rational and integer function o/*x. 

We perceive, therefore, that if it be required to substitute numbers in 
arithmetic progression, as is the case in the resolution of numerical equa 
tions, according to Newton s Method of Divisors, it will suffice to find 
the (m + 1) first results, to form the first, second, &c. differences. The 



ANALYTICAL GEOMETRY. Ixxxi 

mlh difference will have but one term ; as we know it is constant and 
= 1 . 2 . 3 . . . m k h Ir , we can extend the series at pleasure. That of 
the (m l)th differences will then be extended to that of two known 
terms, since it is an arithmetic procession ; that of the (m 2)th differ 
ences will, in its turn, be extended ; and so on of the rest. 

This is perceptible in the preceding example, and also in this; viz. 





x = 





. 1 


.2. 


3 


3d Diff. 6 . 


6 . 


6 


. 6 


. 6 . 


6 


Series 


1 


.-1 


.1. 


13 


2nd . . 4 . 


10. 


16 


. 22 


. 28 . 


34 


1st. 


... 




-2 


.2. 


12 


1st . 2 . 


2 . 


12 


. 28 


. 50. 


78 


2nd 


. . 






3. 


10 


Results 1 . 


j 


1 


. 13 


. 41 . 


91 


3d 


... 








6 


For x . 


1 . 


2 


. 3 


. 4 . 


5 



These series are deduced from that which is constant 

6.6.6.6... 

and from the initial term already found for each of them : any term is 
derived by adding the two terms on the left which immediately precede it. 
They may also be continued in the contrary direction, in order to obtain 
the results of x r= 1, 2, 3, &c. 

In resolving an equation it is not necessary to make the series of results 
extend farther than the term where we ought only to meet with numbers 
of the same sign, which is the case when all the terms of any column are 
positive on the right, and alternate in the opposite direction ; for the 
additions and subtractions by which the series are extended as required, 
preserve constantly the same signs in the results. We learn, therefore, 
by this method, the limits of the roots of an equation, whether they be 
positive or negative. 

Let y x denote the function of x which is the general term, viz. the 
x + 1th, of a proposed series 



y y 2 + yi + . . . y x + 

which is formed by making 

x 0, 1, 2, 3 ... 

For example, y 5 will designate that x has been made 5, or, with re 
gard to the place of the terms, that there are 5 before it (in the last ex 
ample this is 91). Then 

yi yo = A yo y* yi = A yi ya y s = ys 

Ay Ay = A 2 y , Ay* A* yi = A^y, , A y 3 _ A y 2 = A*y 8 . . . 
A 2 y, A*y = A3y , A*y 2 A*y, = A y, , A*y 3 A*y 2 = A 3 y 2 . . . 
&C. 



Ixxxii . INTRODUCTION. 

nix] generally we have 

y x y x _i = Ay x _! 

Ay x Ay x _ j = A-y x _ i 
A 2 y x A*y x _, = A 3 y x _! 
&c. 

Now let us form the differences of any series a, b, r, d . . . in this 
manner. Make 

. b = c + a 
c = b + b 
d = c + c 

&c. 

b - a + a" 
c = b + b" 
d = c + c" 
&c. 

b" = a" + a" 
c" = b" + b" 
d" = c" + c!" 

&c. 

and so on continually. Then eliminating b, b , c, c , &c. from the first 
set of equations, we get 

b = a + a 

c a + 2 a + a" 

d = a + 3 a + 3 a" + a" 

e = a + 4 a + 6 a" + 4 a " + a"" 

f = a + 5 a + 10 a" + &c. 

ic. 

Also we have 

a = b a 
a" = c 2 b + a 
of" =d 3c + 3b a 
&c. 

But the letters a , a", a ", &c. are nothing else than A y ( ,, A ? y , A 3 y . . . 
a, b, c . . . being y c , y b y. 2 . . . , consequently 

y, = y () + A y 
y 2 = Jo + 2 A y + A ? y 
y 3 = y + 3 A y + 3 A 2 yo + ^o 
&c. 



ANALYTICAL GEOMETRY. Ixxxiii 

And 

A yo = y, y 

A2 yo = y 2 2 y, + y 

A3 yo = y s - 3 ya + 3 yj y 

A yo = 74 4 y 3 + 6 y a + 4 y! -{- y c 

&c. 
Hence, generally, we have 



= yo 



1 n 2 



These equations, which are of great importance, give the general term 
of any series, from knowing its first term and the first term of all the 
orders of differences ; and also the first term of the series of nth differ 
ences, from knowing all the terms of the series y , yi, y., . . . 

To apply the former to the example in p. (81), we have 

A v 2 

jo ~ 

A 3 y = 6 

whence 

y x =l 2x + 2x(x l)-fx(x l)(x 2) = x 3 x 2 2x-f-l 
The equations (A), (B) will be better remembered by observing that 

yo \y *) 

provided that in the developements of these powers, we mean by the 
exponents of A y fl , the orders of differences, and by those of y the place 
in the series. 

It has been shown that a, b, c, d . . . may be the values of y x , when 
those of x are the progressional numbers 

ra, m + h, m + 2 li . . . m + i h 
that is 

a = y m , b = y m + h , c - c. 

In the equation (A), we may, therefore, put y m + ih for y x , y m fory , A y m 
fr A y > &c. and, finally, the coefficients of the i th power. Make i h = z, 
and write A, A 2 ... for A y m , A y m . . . and we shall get 

.Vm z = } m + - A ~ + M 2 ^ 1 ) ^ ) Z (Z - h) (Z - 2 h) A 3 



l xxx iv INTRODUCTION. 

This equation will give y x when x = m + z, z being either integer or 
fractional. We get from the proposed series the differences of all orders, 
and the initial terms represented by A, A 2 , &c. 

But in order to apply this, formula, so that it may be limited, we must 
arrive at constant differences ; or, at least, this must be the case if we 
would have A, A 2 ... decreasing in value so as to form a converging 
series : the developement then gives an approximate value of a term cor 
responding to , 

x = m + z; 

it being understood that the factors of A do not increase so as to destroy 
this convergency, a circumstance which prevents z from surpassing a 
certain limit. 

For example, if the radius of a circle is 1000, 

the arc of 60 has a chord 1000,0 _ . fl 

65o 1074,6 A - I* A = - 2,0 

70 1147,2 _ 23 

75 1217,5 

Since the difference is nearly constant from 60 to 75, to this extent 
of the arc we may employ the equation (C); making h = 5, we get for 
the quantity to be added to y = 1090, this 

}.74,6. z / s z (z 5) = 15,12. z 0,04. z 2 

So that, by taking z = 1, 2, 3.. . then adding 1000, we shall obtain the 

chords of 61, 62, 63 ; in the same manner, making z the necessary 

fraction, we shall get the chord of any arc whatever, that is intermediate 

to those, and to the limits GO and 75. It will be better, however, when 

it is necessary thus to employ great numbers for z, to change these limits. 

Let us now take 

lo<*. 3100 = y = 4913617 

m A. = 13987 
log. 3110 = 4927604 A * = 45 

13942 
log. 3120 = 4941546 _ 45 

13897 

log. 3130 = 4955443 

We shall here consider the decimal part of the logarithm as being an 
integer. By making h = 10, we get, for the part to be added to log. 

3100, this 

1400,95 x z 0, 225 X z 2 . 
To get the logarithms of 3101, 3102, 3103, &c. we make 

z = 1,2, 3....; 
and in like manner, if we wish for the log. 3107, 58, we must make 



ANALYTICAL GEOMETRY. Ixxxv 

z 7, 58, whence the quantity to be added to the logarithm of 3100 is 
10606. Hence 

log. 310768 = 5,4924223. 

The preceding methods may be usefully employed to abridge the 
labour of calculating tables of logarithms, tables of sines, chords, &c. 
Another use which we shall now consider, is that of inserting the inter 
mediate terms in a given series, of which two distant terms are given. 
This is called 



INTERPOLATION. 



It is completely resolved by the equation (C). 

When it happens that A 2 = 0, or is very small, the series reduces to 

z L y A 

TT 

whence we learn that the results have a difference which increases propor 
tionally to z. 

When A 2 is constant, which happens more frequently, by changing z 
into z + 1 in (C), and subtracting, we have the genera] value of the first 
difference of the new interpolated series ; viz. 

First difference A = - + 2 z ~ h + 1 A 
h 2 h 2 

Second difference A" =: ,---. 

If we wish to insert u terms between those of a given series, we must 
make 

h = n + 1 ; 
then making z = 0, we get the initial term of the differences 

A. 2 



A 



(11 + 

A . 



we calculate first A", then A ; the initial term A will serve to compose 
the series of first differences of the interpolated series, (A" is the constant 
difference of it); and then finally the other terms are obtained by simple 
additions. 

If we wish in the preceding example to find the log. cf 3101, 



INTRODUCTION. 

3102, 3103 ... we shall interpolate 9 numbers between those which arc 
given : whence 

u ^ 9 

A" = 0,45 

A = 1400,725. 

We first form the arithmetical progression whose first term is A , and 
0,45 for the constant. The first differences are 

1400,725; 1400,725; 1399,375; 1398,925, &c. 

Successive additions, beginning with log. 3100, will give the consecutive 
logarithms required. 

Suppose we have observed a physical phenomenon every twelve hours, 
and that the results ascertained by such observations have been 

For hours . . . 78 _ 

12 ... 300 z A 2 = 144 

24 ... 666 

36 ... 1176 510 144. 

&c. 

If we are desirous of knowing the state corresponding to 4 h , 8 h , 12 h , 
&c., we must interpolate two terms; whence 

ti = z, A" = 16, A = 58 

composing the arithmetic progression whose first term is 58, and common 
difference 16, we shall have the first differences of the new series, and 
then what follow 

First differences 58, 74, 90, 106, 122, 138 ... 

Series 78, 136, 210, 300, 406, 528, 64G , . . 

A O h , 4 h , 8 h , 16 h 20 h , 24 ". 

The supposition of the second differences being constant, applies almost 
to all cases, because we may choose intervals of time which shall favour 
such an hypothesis. This method is of great use in astronomy; and 
even when observation or calculation gives results whose second differ 
ences are irregular, we impute the defect to errors which we correct by 
establishing a greater degree of regularity. 

Astronomical, and geodesical tables are formed on these principles. 
AVe calculate directly different terms, which we take so near that their 
first or second differences may be constant ; then we interpolate to obtain 
the intermediate numbers. 

Thus, when a converging series gives the value of y by aid of that of a 
variable x ; instead of calculating y for each known value of x, when the 
formula is of frequent use, we determine the results y for the continually 



ANALYTICAL GEOMETRY. 

increasing values of x, in such a manner that y shall always be nearly of 
the same value : we then write in the form of a table every value by the 
side of that of x, which we call the argument of this table. For the 
numbers x which are intermediate to them, y is given by simple proposi 
tions, and by inspection alone we then find the results icqaired. 

When the series has two variables, or arguments x and z, the values 
ofy are disposed in a table by a sort of double entry ; taking for coordi 
nates x and z, the result is thus obtained. For example, having made 
z = 1, we range upon the first line all the values ofy corresponding to 

x = 1, > , 3...; 

we then put upon the second line which z z gives ; in a third line those 
\vhich z = 3 gives, and so on. To obtain the result which corresponds to 

x = 3, z = 5 

we stop at the case which, in the third column, occupies the fifth place. 
The intermediate values are found analogously to what has been already 
shown. 

So far we have supposed x to increase continually by the same differ 
ence. If this is not the case and we know the results 

y = a, b, c, d . . . 
which are due to any suppositions 



X = a 



we may either use the theory which makes a parabolic curve pass through 
a series of given points, or we may adopt the following: 
By means of the known corresponding values 

a, a ; b |8 ; &c. 
we form the consecutive functions 



b a 



c-fa 

y fl 

d c 

6 -, 

B = A ~ A 
7 a 

B = AlZZ_A 



A - 



A. = 



Ikcj 

/4 



bcxxviii INTRODUCTION. 

r 1 -- BI B 



&c. 

C 



v a 
and so on. 

By elimination we easily get 
b = a + A ((3 a) 

c = a + A (7 a) + B( 7 a) (7 /3) 

d = a + A (3 ) + B(3 a) (3/3) + C(S a)(3 /3) (3 7) 
&c. 

and generally 

y x = a+A(x a) + B(x a )( x _/3) + C (x a) (x /3) (x 7) + &c. 
We must seek therefore the first differences amongst the results 

a, b, c . . . 
and divide by the differences of 

a, ft 7 ... 
which will give 

A, A 19 A 2 , &c. 

proceeding in the same manner with these numbers, we get 

B, Bj, Ba, &c. 
which in like manner give 

C, C,, Cs, &c. 

and, finally substituting, we get the general term required. 
By actually multiplying, the expression assumes the form 

a + a x + a x 2 ^-... 

of every rational and integer polynomial, which is the same as when we 
neglect the superior differences. 



The chord of 60 = rad.=rlOOO 



=1035 



65. 10 = 



A =15 

Aj = 14,82 
A 2 = 14,61 



B =0,035 
B 1 = 0,031 



69. =1133 
We have 

a = 0, /3 = 21, 7 = 5^, 8 = 9. 
We may neglect the third differences and put 

y x = 100 + 15,082 x 0,035 x 2 . 
Considering every function of x, y x , as being the general term of the 



series which gives 



x = m, m + b, m + 2 h, &c. 



ANALYTICAL GEOMETRY. Ixxxix 

if we take the differences of these results, to obtain a new series, the 
general term will be what is called the first difference of the proposed 
function y x which Is represented by A y x . Thus we obtain this difference 
by changing x into x + h in y x and taking y x from the result ; the re 
mainder will give the series of first differences by making 

x = m, m + h, m -f 2 h, &c. 
Thus if 



(x 



J x " a + x + h a + x 

It will remain to reduce this expression, or to develope it according to 
the increasing powers of h. 

Taylor s theorem gives generally (vol. I.) 
d y d 2 y h 2 

To obtain the second difference we must operate upon A v x as upon <(he 
proposed y x , and so on for the third, fourth, &c. differences. 

INTEGRATION OF FINITE DIFFERENCES. 

Integration here means the method of finding the quantity whose dif 
ference is the proposed quantity ; that is to say the general term y x of a 

Jin? ym + h> ym + 2h) & c - 

from knowing that of the series of a difference of any known order. This 
operation is indicated by the symbol 2. 
For example 

2 (3x 2 + x 2) 
ought to indicate that here 

h = 1. 

A function y x generates a series by making 
x = 0, 1, 2, 3 ... 
the first differences which here ensue, form another series of which 

3 x 2 + x 2 
is the general term, and it is 

2, 2, 12, 28 ... 

By integrating we here propose to find y x such, that putting x -f- 1 for 
x, and subtracting, the remainder shall be 
3 x " + x 2. 



xc INTRODUCTION. 

It is easy to perceive that, first the symbols 2 and A destroy one another 
as do f and d; thus 

2 A f x = f x, 
Secondly, that 

A (a y) = a A y 
gives 

2 a y a 2 y. 
Thirdly, that as 

A (A t B u) = A A t n A u 
so is 

2 (A t B u) = A 2 t B 2 u, 
t and u being the functions of x. 

The problem of determining y x by its first difference does not contain 
data sufficient completely to resolve it ; for in order to recompose the 
series derived from y x in beginning with 

2, 2, 12, 28, &c. 
we must make the first term 

.Vo = a 
and by successive additions, we shall find 

a, a 2, a + 2, a + 12, &c. 
in v/hich a remains arbitrary. 

Kvery integral may be considered as comprised in the equation (A) 
p. 83 ; for by taking 

x = 0, 1, 2, 3 . . . 

in the first difference given in terms of x, we shall form the series of first 
differences ; subtracting these successively, we shall have the second dif 
ferences ; then in like manner, we shall get the third, and fourth differ 
ences. The initial term of these series will be 

A y u , A -y . . . 

and these values substituted in y x will give y x . Thus, in the example 
above, which is only that of page (81) when a = 1, we have 

A y = 2, A 2 y = 4, A 3 y = 6, A * y Q = 0, &c. ; 
which give 

y x = y 2 x x 2 + x 3 . 

Generally, the first term y of the equation (A) is an arbitrary constant, 
which is to be added to the integral. If the given function is a second 
difference, we must by a first integration reascend to the first difference 
and thence by another step to y x ; thus we shall have two arbitrary con 
stants ; and in fact, the equation (A) still gives y x by finding A s , A 3 , the 



ANALYTICAL GEOMETRY. xci 

only difference in the matter being that y and A y are arbitrary. And 
so on for the superior orders. 

Let us now find 2 x m , the exponent m being integer and positive. 
Represent this developement by 

2 x m = p x + qx b + rx c -f- &c. 

a, b, c, &c. being decreasing exponents, which as well as the coefficients 
p, q, &c. must be determined. Take the first difference, by suppressing 
2 in the first member, then changing x into x + h in the second member 
and subtracting. Limiting ourselves to the two first terms, we get 

o o y o 

x m = pahx 3 - 1 + pa(a I)h 2 x a - 2 + . . . qbh x"- 1 + . .. 
But in order that the identity may be established the exponents ought 
to give 

a ] = m 
a 2 = b 1 
whence 

a = m + 1, b rr m. 
Moreover the coefficients give 

I = p a h, % p a (a 1 ) h q b ; 
whence 

P = (ni + 1) h q = ~ * 

As to the other terms, it is evident, that the exponents are all integer 
and positive ; and we may easily perceive that they fail in the alternate 
terms. Make therefore 

2x m = px rn + 1 x m + ax m - T -f ,Sx m - 3 -f 7 x m ~ 5 + . .. 
and determine , j3, y ... &c. 

Take, asbefore, the first difference by putting x + h for x, and sub 
tracting : and first transferring 

o o 

Pv m + 1 __ L v m 
X 2 X , 

we find that the first member, by reason of 
p h (m +!) = !, 
reduces to 



_ 
2.3 4- 2.5 6 2.7 

To abridge the operation, we omit here the alternate terms of the deve 
lopement; and we designate by 

1, in, A , A , &c. 
the coefficients of the binomial. 

Making the same calculations upon 

a x 1 "- 1 + /3 x ln - 3 + &c. 



xcii INTRODUCTION. 

we shall have, with the same respective powers of x and of h, 
(m-l) a +(m-l).!=2.m^ + (m-l}. ^.. m 

+ (m-3)0+(m._S).2^p*...!IL=-?,3 + .. 

A O 

+ (m 4) 7 +.. 

Comparing them term by term, we easily derive 

m 

A" 



" ~~ 2.3.4.5 

. A//// 
7 ~~ 6.6.7 
&c. 
whence finally we get 



+ A""ch 5 x m - 5 + A vi dh 7 x m - 7 +...(D) 

This developement has for its coefficients those of the binomial, taken 
from two to two, multiplied by certain numerical factors a, b, c . . ., which 
are called the numbers of Bernoulli, because James Bernoulli first deter 
mined them. These factors are of great and frequent use in the theory 
of series ; we shall give an easy method of finding them presently. These 
are their values 

J^ 
= 12 

b = - 



120 
1 



~ 252 



240 
1 
6 ~ 132 

691 



f = 



32780 
I 
12 



h = - 



8160 
. _ 43867 
~ 14364 
&C. 



ANALYTICAL GEOMETRY. 

which it will be worth the trouble fully to commit to memory. 

From the above we conclude that to obtain 2 x m , m being any number, 
integer and positive, we must besides the two first terms 

x m + 1 x m 

(m + 1) h 2~ 
also take the developernent of 

(x + h) m 

reject the odd terms, the first, third, fifth, &c. and multiply the retained 
terms respectively by 

a, b, c . . . 

Now x and h have even exponents only when m is odd and reciprocally : 
so that we must reject the last term h m when it falls in a useless situation ; 
the number of terms is | m + 2 when m is even, and it is | (m + 3) when 
m is odd ; that is to say, it is the same for two consecutive values of m. 
Required the integral ofx 10 . 
Besides 

x 11 

_ ___ 1 ,,10 

11 h 

we must develope (x + h) % retaining the second, fourth, sixth, &c. terms 
and we shall have 

10x 9 ah+ 120x 7 bh 3 + 252x 5 ch 5 + &c. 
1 herefore 






In the same manner we obtain 



2x o _ * 

h 



2X 1 - 


X* 


x 




2 h 


2 


v X 5 


X 3 


x 2 h x 




3~h 


2 + 6 


2 X 3 - 


x 4 


x 3 hx 2 




4h 


2 4 


2 X 4 - 


x 5 


x 4 hx 3 h 3 x 




5 h 


4 3 30 


V v 5 

A. ^_ 


x 6 


x 5 5hx 4 h 3 x 2 


6h 


2 12 12 


2 X fi - 


x 7 


x 6 h x 5 h 3 x l - h 5 x 




7 h 


2 2 6 42 


2 X 7 = 


x 8 
8li 


x 7 7 h x c 7h 3 x 4 h 5 
" 2 12 -2T~ + "I 



xciv INTRODUCTION 



- 8 _ __. _ _ , _ _ 

~ y h " 2 + 3 15 9 " 30" 

9-^-^4. 3 h x _ 7 h 3 x ti 5 x 4 __ 3 h 7 x 2 
= 10 h ~" 2 4 10 2 20 

x 11 

5 x 10 = - -- &c. as before, 
11 h 

&c. 

We shall now give an easy method of determining the Numler of 
Bernoulli a, b, c . . . In the equation (D) make 

X = h = 1; 

2 x m is the general term of the series whose first difference is x . We 
shall here consider 2. x = ], and the corresponding series which is that 
of the natural numbers 

0, 1, 2, 3 ... 
Take zero for the first member and transpose 

JL i 

m + 1 " 
which equals 



I m 
Then we et 



= a m 



+ b A" + c A Iv + cl A * -f . . . + k m. 



2 (m+ 
By making m = 2, the second member is reduced to am, which gives 



Making m = 4, we get 

3 = 4 a + b K" 



10 



m 1 m 2 . 
4 a + m . . b 



4 a + 4 b 

= f + 4 b. 

Whence 

b - _L 

120* 

Again, makiug m = 6, we get 
5 
= 6 a + b A" + c A 

= 6a+ 20 b +6c 

= i -i- + o c 



ANALYTICAL GEOMETRY. xcv 

which gives 

s\ < , _ . 
" 252 

and proceeding thus by making 

m = 2, 4, 6, 8, &c. 

we obtain at each step a new equation which has one term more than the 
preceding one, which last terms, viz. 

2 a, 4 b, 6 c, . . . m k 
will hence successively be found, and consequently, 

a, b, c . . . k. 
Take the difference of the product 

y x = (x h)x (x + h) (x + 2h)...(x+ih), 
by x + h for x and subtracting ; it gives 

A y x = x (x + h) (x + 2 h) ... (x + i h) x (i + 2) h; 
dividing by the last constant factor, integrating, and substituting for y x 
its value, we get 

2x (x + h) (x+ 2h)...(x + ih) 



Xx ( x 



This equation gives the integral of a product of factors in arithmetic 
progression. 

Taking the difference of the second member, we veiify the equation 

v _ 1 _ = __ - _ ; 

x (x + h) (x + 2 h) . . .(x + i h) i h x (x + h) . . . [x + (i 1) h} 

which gives the integral of any inverse product 
Required the integral of a.*. 
Let 

v - n X 

}x 

Then 

Ay* = a x (a h 1) 
whence 

y x = 2a x (u" I) = a x ; 
consequently 

a x 

5 a x = r - - + constant. 
a h 1 

Required the integrals qfs m. x, cos. x. 
Since 

cos. B cos. A = 2 sin. % (A + B). sin. (A B) 
A cos. x = cos. (x + h) cos. x 

hx h 



= 2 sin. (x + - -) sin. 



xcvi INTRODUCTION. 

Integrating and changing x + - - into z, we have 

m 

( *) 

2 sm. z = cos. -f- constant. 

h 

In the same way we find 



2 cos. z = { f- constant. 

h 
2 sin.- 

When we wish to integrate the powers of sines and cosines, we trans 
form them into sines and cosines of multiple arcs, and we get terms of 
the form 

A sin. q x, A cos. q x~ 
Making 

q x rr x 

the integration is performed as above. 
lieguired the integral of a product, viz. 
Assume 

2(uz) = u2z + t 

u, z and t being all functions of x, t being the only unknown one. By 
changing x into x + h in 

u 2 z + t 

u becomes u + A u, z becomes z + A z, &c. and we have 
u2z+uz + Au2(z + Az) + t+At; 
substituting from this the second member 

u 2 z + t, 
we obtain the difference, or u z ; whence results the equation 

= Au2(z + A Z ) + At 
which gives 

t = 2 A u 2 (z + A z )}. 
Therefore 

2 (u z) = U 2 Z 2 {A u . 2 (Z + A z)] 
which is analogous to integrating by parts in differential functions. 

There are but few functions of which we can find the finite integral ; 
when we cannot integrate them exactly, we must have recourse to series. 

Taylor s theorem gives us 

dy. , d 2 y h s 
A y x = , h + -r-v -^- + &c. 
J * dx dx 2 ^ 



ANALYTICAL GEOMETRY. 



by supposition. Hence 

y x = h 2 y + ~ 2 y" + &c. 
Considering y as a given function of x, viz. z, we have 

y = * 

y/// __ 7 n 

&c. 
and 

y x = /y dx = /zdx 
whence 

h 2 
/z d x = h 2 z + 2 z + &c. 

2f 

which gives 

2 z = h- 1 /"z d x 4 2 z 7- h 2 2 z " &c - 

o 

This equation gives 2 z, when we know z , 2 z , &c. Take the dif 
ferentials of the two numbers. That of the first 2 z will give, when di 
vided by d x, 2 z . Hence we get 2 z", then 2 z" , &c. ; and even without 
making the calculations, it is easy to see, that the result of the substitution 
of these values, will be of the form 

2 z = h- /z d x + A z + B h z + C h 2 z" + &c. 
It remains to determine the factors A, B, C, &c. But if 

z = x m 
we get 

/z d x, z , z", &c. 

and substituting, we obtain a series which should be identical with the 
equation (D), and consequently defective of the powers m 2, m 4, 
so that we shall have 

_/-zdx z a h z b h z " , cW"" dhV""" , 
h 2~ H ~T~ ~TT 2.3.4 2... 6 

a, b, c, &c. being the numbers of Bernoulli. 
For example, if 

z = 1 x 

yix.dx = x 1 x x 
z = x- 1 
z" = &c. 



xcviii INTRODUCTION 

consequently 

2lx = C-fxlx x lx + a x- 1 + b x~ 3 + c x-/ + Sec. 
The series 

a, b, c . . . k, 1, 
having for first differences 

i h c I 

B 9 V 9 v K 

we have 

b = a + a 
c = b + b 
(1 = c + c 

ate. 

i = k + k 

equations whose sum is 

1 = a + a + b + c + . . . k . 

If the numbers a , b , c , &c. are known, we may consider them as being 
the first differences of another series a, b, c, &c. since it is easy to com 
pose the latter by means of the first, and the first term a. By definition 
we know that any term whatever 1 , taken in the given series a , b , c , &c. 
is nothing else than A 1, for 1 = m 1 ; integrating 

T = A 1 

we have 

21 =2 1 

or 

2 1 = a + b + c . . . + k , 

supposing the initial a is comprised in the constant due to the integra 
tion. Consequently 

The integral of any term whatever of a series^ we obtain the sum of all 
the terms that precede it, and have 

2 y x = y + yi + y + y * - 1. 

In order to get the sum of a series, we must add y x to the integral ; or 
which is the same, in it must change x into x + 1, before we integrate. 
The arbitrary constant is determined by finding the value of the sum y 
when 

x = 1. 

We know therefore how to Jind the summing term of every series whose 
general term is known in a rational and integer function ofx. 
Let 

y x = A x m B x n + C 
m ;and n being positive and integer, and we have 

A2x ra B 2 x" + C 2 x 



ANALYTICAL GEOMETRY. xcix 

for the sum of the terms as far as y x exclusively. This integral beino 
once found by equation D, we shall change x into x + 1, and determine 
the constant agreeably. 
For example, let 

y= x(2x-l); 
changing x into Z + 1, and integrating the result, we shall find 

A -.- 3 I O 9 

2 2x 3 + 3 2x+ 2 X = 



= x . 



2.3 
x + 1 4 x ] 



2 3 

there being no constant, because when x = 0, the sum = 0. 
The series 

l m , 2 m , 3 m ... 

of the m th powers of the natural numbers is found by takimr 2 x m (equa 
tion D); but we must add afterwards the x th term which is x m ; that is to 
say, it is sufficient to change x m , the second term of the equation 
(D), into x m ; it then remains to determine the constant from the term 
we commence from. 
For example, to find 

S = 1 + 2* + 3 2 + 4 + .,.x 
we find 2 x 2 , changing the sign of the second term, and we have 

x 3 x 2 x x+ 1 2x + 1 

S --3 +-2 + 6- = X -3-- iH 

the constant is 0, because the sum is when x = 0. But if we wish to 
find the sum 

S = (n + I) 2 + (n + 2) 2 + ...x* 
S = 0, whence x = n 1, and the constant is 

n 1 2 n 1 
2 ~3 > 

which of course must be added to the former ; thus giving 

S = (n + 1)* + ( n + 2) +... x * 

x + 1 2x -f 1 n 1 2 n 1 

3 2 ~2~ -- 3~ 

= - - X {x.( x + 1). (2x + l)_ n .(n 1) (2 n I) 



This theory applies to the summation ofjgurate numbers, of the dif 
ferent orders : 



c INTRODUCTION. 

First order, 1.1.1.1.1. 1 . 1 , &c. 
Second order, 1.2.3.4.5. 6 . 7 , &c. 
Third order, 1.3. 6 . 10 . 15 . 21 . 28 , &c. 
Fourth order, 1 . 4 . 10 . 20 . 35 . 56 . 84 , &c. 
Fifth order, 1 . 5 . 15 . 35 . 70 . 126 . 210, &c. 

and so on. 

The law which every term follows being the sum of the one immediate 
y over it added to the preceding one. The general terms are 

First, 1 

Second, x 

Third, X (X 2 +1) 

r . x (x + 1) (x + 2) 
Fourth, v oV 



D tn x.(x+ 1) (x + 2)...x + p 2 

1 . 2 . 3 . . . p 1 
To sum the Pyramidal numbers, we nave 

S = 1 + 4 + 10 + 20 + &c. 
Now the general or x th term in this is 

y x = -1 . x (x + 1) (x + 2). 
But we find for the (x 1)* term of numbers of the next order 

2l ( x !) x ( x + 1) (x + 2) ; 

finally changing x into x + 1, we have for the required form 
S = ^-x.(x + l)(x + 2)(x +3). 

Since S = 1, when x = 1, we have 

1 = 1 + constant, consequently 
.\ constant = 0. 

Hence it appears that the sum of x terms of the fourth order, is the 
x tb term or general term of the fifth order, and vice versa ; and in like 
manner, it may be shown that the x th term of the (n + l) th order is the 
sum of x terms of the n th order. 

Inverse Jigurate numbers are fractions which have 1 for the numerator, 
and a figurate series for the denominator. Hence the x th term of the p th 
order is 

1 . 2 . 3 . . . (p _!)_ 
x (x + 1).. .x + p 2 



ANALYTICAL GEOMETRY. oi 



and the integral of this is 



(p 2)x(x +l)...(x + p 3) 

Changing x into x+1, then determining the constant by making 
x = 0, which gives the sum = 0, we shall have 

p 1. 

and the sum of the x first terms of this general series is 

p 1 1.2.3...(p 1) 

p _ 2 (p 2) (x + 1) (x + 2) . . . (x + p 2)* 

In this formula make 

p - 3, 4, 5 ... 
and we shall get 

1 4. * 1 ! 4. 1 4. 1 - 2 - 2 2 

1 ~ 3 " ~ 6 " r 10 "* x (x + 1) 1 x+1 

_!_ I !_ 1 1.2.3 _ 3 3 

1 " 4 * 10 + 20 + " x (x + J ) (x + 2) " 2 (x + 1) (x + 2) 
1 1 , J_ _L 1 .2.3.4 _ 4 2.4 

T + 5 + T0 "*" 35 + *"x(x+l)(x+2) (x + 3) ~ 3" (x+ J)...(x + 3) 
1 1^ 1 1.2.3.4.5 5 2.3.5 

T + 6 + 21 + 56 + *"x(x+l)...(x + 4) ~ 4 (x+1) . ..(x + 4) 



and so on. To obtain the whole sum of these series continued to infinity, 
we must make 

X = CD 

which gives for the sum required the general value 

P-l 

P -2 

which in the above particular cases, becomes 

2345 



1 2 3 4 &C> 



To sum the series 



sin. a + sin. (a + h) + sm . (a + 2 h) + . . . sin. (a + x 1 h) 
we have 

cos. (a + h x -- -J 

? sin. (a + x h) = C -- j - 

2sin.J 

changing x into x + 1, and determining C by the condition that x = J 
makes the sum = zero, we find for the summing-term. 

cos. 



. (a ^-) cos. (a + h x + 



cii INTRODUCTION. 

or 

sin. 



. / , h x . h (x + 1) 
sin. (a + xj sin. ^ 



In a similar manner, if we wish to sum the series 

cos. a + cos. (a + h) -f- cos. (a -f 2 h) + . . . cos. (a + x 1 h} 
we easily find the summing-term to be 

sin. (a. ^\ sin. ( a + h x -f -- \ 

\ - d / \ lil 

2 sin. A 

or 

h ^ h (x 4. 1) 

__ . 

nn. 



cos. t - , g 



A COMMENTARY 



ON 



NEWTON S PRINCIPIA, 



SUPPLEMENT 

TO 

SECTION XL 

460 PROP. LVII, depends upon Cor. 4 to the Laws of Motion, 
which is 

If any number of bodies mutually attract each other, their center of gra 
vity will either remain at rest or will move uniformly in a straight line. 
First let us prove this for two bodies. 
Let them be referred to a fixed point by the rectangular coordinates 

*> y ; x , y , 

and let their masses be 

(* /* . 

Also let their distance be ? , and f () denote the law according to which 
they attract each other. 

Then 



will be their respective actions, and resolving these parallel to the axes of 
abscissas and ordinates, we have (46) 



VOL. II. 




A COMMENTARY ON [SECT. XL 



Hence multiplying equations (1) by ^ and those marked (2) by ft, and 
adding, &c. we get 



dt 



= 0, 



and 



dt 2 
and integrating 

d x , d x 



dt dt 

Now if the coordinates of the center of gravity be denoted by 

x, y, 

we have by Statics 

- /a x jf x 7 



_ 



+ 



+ ft 

d x _ 1 / d x , dx\ 



d t /i + fjf \ d t d t / 

and 

^ y 

dt" = . 
But 

d x d y 
dl "dT 

represent the velocity of the center of gravity resolved parallel to the axes 
of coordinates, and these resolved parts have been shown to be constant 
Hence it easily appears by composition of motion, that the actual velocity 
of the center of gravity is uniform, and also that it moves in a straight 
line, viz. in that produced which is the diagonal of the rectangular par 
allelogram whose two sides are d x, d y. 
If 

c = 0, c = 
then the center of gravity remains quiescent. 



BOOK!.] NEWTON S PRINCIPIA. 3 

461 The general proposition is similarly demonstrated, thus. 
Let the bodies whose masses 

I* , p", /", &c. 
be referred to three rectangular axes, issuing from a fixed point by the 

coordinates 

x v 7 
A ? y 5 z 



/ 

y 

&c. 



x ", y" , -z!" 



Also let 

^i j 2 be the distance of //, // 



&C. &C. 

and suppose the law of attraction to be denoted by 

f -(*i,2) f(fi,s)> f (fo,s) & c - 
Now resolvin the attractions or forces 



&c. 
parallel to the axes, and collecting the parts we get 

d 2 x x x" , ,, ff , .x - 

= /* I (?i o ) -4- U> I ( ?i -j) 



&C. 
&c- 



t O, O 0.1 Q 

jl, J j , 9 

&c. = &c. 



Hence multiplying the first of the above equations by [jf t the second by 
a 7 , and so on, and adding, we get 

gM^ + ^d x" + it!" d g x " + &c. _ . 

~dT 2 ~ 

Again, since it is a matter of perfect indifference whether we collect the 
forces parallel to the other axes or this ; or since all the circumstances are 
similar with regard to these independent axes, the results arising from 
similar operations must be similar, and we therefore have also 
fif d 2 y + tt," d 2 y" + f* " d 8 y w + &c. _ 

dt 8 

d 2 z" + " d g T!" + &c. _ 
dt ~~- 

A 2 



4 A COMMENTARY ON [SECT. XI. 

Hence by integration 

, dx , ,.dx" , ^dx " , 

""d7 + /A -dT +/i "dT + &c = c 

d v d v" d v" 

/ -^T + *"TT + / ""dT + &c = <; 

1 rl / rl / <1 7 ///( 

. U Z .. U t, ... U. 1 .. 

* dl +fJ> -dT + " -dF +&c =c 

But x, y, z denoting the coordinates of the center of gravity, by statics 
we have 

- - / * + V>" *" + V- " * " + &c. 

tf + p + p" + &C. 

_ p f + ^ y /r + ^ w y /7/ + &c. 

tf + X + <"" + &c. 
_ _ p z + ^ z ^ + (,/ i>" + &c. 

p + p," + ^ + & c . 
and hence by taking the differentials, &c. we get 

dx c 

d t " p! + fjf + v!" + &c. 

d y (^ 

^" 



d t /* + n" + u!" + &c. 

that is, the velocity of the center of gravity resolved parallel to any three 
rectangular axes is constant. Hence by composition of motion the actual 
velocity of the center of gravity is constant and uniform, and it easily ap 
pears also that its path is a straight line, scil. the diagonal of the rectan 
gular pai allelopiped whose sides are d x, d y, d z. 

462. We will now give another demonstration of Prop. LXI. or that 
Of two bodies the motion of each about the center of gravity, is the same 
as if that center was the center of force, and, the law of force the same as 
that of their mutual attractions. 

Supposing the coordinates of the two bodies referred to the center of 
gravity to be 

we have 

x = x + x, ^ x = x + x,, 

y=~y + y/ j y = y + y/, 

Hence since 

d x d y 
eft dT 



BOOK I.] NEWTON S PRINCIPIA. 

are constant as it has been shown, and therefore 



" 



d 2 x dy 

. 



, _ . _ . 

dt 2 cit 

we have 

<Px _d g x/ 
dt 2 "TF 
cPy _ d* y/ 
dt 2 " dt" 

and we therefore get (46) 



But by the property of the center of gravity 



being the distance of /* from the center of gravity. We also have 



f T" 

Hence by substitution the equations become 



Similarly we should find 



and 



Hence if the force represented by 



were placed in the center of gravity, it would cause /" to move about it as 
a fixed point; and if 



were there residing, it would cause ^ to centripetate in h"ke manner. 
Moreover if 



A 3 



6 A COMMENTARY ON [SECT. XI 

then these forces vary as 

a /n , a n ; 

so that the law of force &c. &c. 

ANOTHER PROOF OF PROP. LXII. 



463. Let p, [i! denote the two bodies. Then since & has no motion 
round G (G being the center of gravity), it will descend in a straight 
line to G. In like manner p will fall to G in a straight line. 

Also since the accelerating forces on p, tf are inversely as /*, p or 
directly as G A, G //, the velocities will follow the same law and corre 
sponding portions of G ^ G tf will be described in the same times ; that 
is, the whole will be described in the same time. Moreover after they 
meet at G, the bodies will go on together with the same constant velocity 
with which G moved before they met. 

Since here 



a. will move towards G as if a force 

^^ 



or 



Hence by the usual methods it will be found that if a be the distance 
at which <. begins to fall, the time to G is 

+ p f ) a 2 v 



^l 2V2 
and if a be the original distance of/* , the time is 

(ft + X) of * cr 

* 2V2 
But 

a : a : : p : p 
therefore these times are equal, which has just been otherwise shown. 



BOOK I.] NEWTON S PRINCIPIA. 7 

ANOTHER PROOF OF PROP. LXII1. 

464. We know from (461) that the center of gravity moves uniformly 
in a straight line; and that (Prop. LVII,) p and fjf will describe about G 
similar figures, p moving as though actuated by the force 



and Q as if by 



Hence the curves described will be similar ellipses, with the center of 
force G in the focus. Also if we knew the original velocities of p and y! 
about G, the ellipse would easily be determined. 

The velocities of /* and [jf at any time are composed of two velocities, 
viz. the progressive one of the center of gravity and that of each round G. 
Hence having given the "whole original velocities required to find the separate 
parts of them, 

is a problem which we will now resolve. 
Let 

V, V 

be the original velocities of /a, //, and suppose their directions to make 
with the straight line p yf the angles 

a, of. 
Also let the velocity of the center of gravity be 

v 
and the direction of its motion to make with p fjJ the angle 

a. 
Moreover let 

v, v 

be the velocities of /*, /// around G and the common inclination of their 
directions to be 

6. 
Now V resolved parallel to p // is 

V cos. . 
But since it is composed of v and of v it will also be 

v cos. a -f- v cos. & 
. . V COS. a = v COS. a -}- v COS. &. 
In like manner we get 

V sin. = v sin. a + v sin. 6. 
A 4 



8 A COMMENTARY ON [SECT. XL 

and also 

V cos. a! v cos. a v cos, 6 

V sin. a! =r v sin. a. v sin. 6. 

Hence multiplying by /*, At , adding and putting 

At v = A 1 v 
we get 

At V COS. a + At V COS. a = (A& + //) V COS. a 
and 

At V sin. a + At V sin. =(/, + ///) 

Squaring these and adding them, we get 

At 2 V 2 -f- y^V 2 + 2A*At VV cos.(a- a ) = 
winch gives 

v= 



(if) V COS. a A 
At ) v sin. a J 



At + X 
By division we also have 

_ ft V sin. a -f- (* V sin. a 
tan. u fj - /r^pr/ / 

^ V cos. a + // V cos. a 

Again, from the first four equations by subtraction we also have 

V cos. a V 7 cos. a (v + v ) cos. 6 = v . ^ , ^ cos. 

^ 

V sin. a V 7 sin. a r= (v + V) sin. = v . ^ , A sin. ^ 

p 

and adding the squares of these 

V* + V /2 2 V V cos. (a aO= v 2 
whence 



v = 7 . ViV 2 + V /2 2 VV cos. (a ) 

f* + 

+ V /2 SVV COS. (a a ) 



and by division 

V sin. a V sin. a 
tan. = ^ - v^ - , . 

V COS. a V 7 COS. a 

Whence are known the velocity and direction of projection of /* about 
G and (by Sect. III. or Com.) the conic section can therefore be found ; 
and combining the motion in this orbit with that of the center of gravity, 
which is given above, we have also that of/*. 

465. Hence since the orbit of fj> round (* is similar to the orbit of 
tt round G, if A be the semi-axis of the ellipse which /* describes round 



BOOK I.] NEWTON S PRINCIPIA. 9 

G, and a that of the ellipse which it describes relatively to /* which is also 
in motion j we shall have 

A : a : : IM/ : /* + /& . 

466. Hence also since an ellipse whose semi-axis is A, is described by 
the force 



we shall have (309) the periodic time, viz. 

T - 2 A ^ff __ 2 g A s (,a + Ap 



2 ff 



V (A* + it, 1 ) 

467. Hence we easily get Prop. LIX. 

For if At were to revolve round /* at rest, its semi-axis would be a, and 
periodic time 



.-. T : T : : V / : V (^ + t* ). 

468. PROP. LX is also hence deducible. For if /* revolve round (if a* 
rest, in an ellipse whose semi-axis is a , we have 



and equating this with T in order to give it the same time about / at rest 
as about & in motion, we have 



.-. a : a : : (/& + ^ ) : ^ . 

ANOTHER PROOF OF PROP. LXIV. 

469. Required the motions of the bodies whose masses are 

ft, //, p, p." , & C . 

and which mutually attract each other with forces varying directly as the 
distance. 

Let the distance of any two of them as p, ,/> be j ; then the force of (i! 
on <j. is 



10 A COMMENTARY ON [SECT. XI. 

and the part resolved parallel to x is 

/ ^ L ~ X . , 

(L 1 s = f (X X ). 

In like manner the force of [*" on p, resolved parallel to x, is 

p" (x x") 

and so on for the rest of the bodies and for their respective forces resolved 
parallel to the other axes of coordinates. 

Hence 

^ = o! (x x ) + ?, (x - x") + &c. 
T=fi (* x) + / (x x") + &c. 



~ = ft (x" - x) + nf (x" x ) + &c. 

&c. = &c. 
which give 

^~ = (p, + f* + p + &c.) x (it, x + ^ x + &c.) 

/X + &C.) 



= (^ + ^ + X + &C.) X" (^ X + ft, X + &C.) 



&c. = &c. 
Or since 

(J, X + (* X 7 + &C. = ([* + iff + &C.) X 

making the coordinates of the center of gravity 

x> y, z", 
we have 



&c. = &c. 
In like manner, we easily get 

?= (" + <* +&c.)(y y) 



BOOK I.] NEWTON S PRINCIPI A. 11 



&c. = &c. 
and also 



f = (,* + v + &c.) (z ~z) 



&c. = &c. 
Again, 

x x , y y , z z 
x x~,y y, z z 

&c. &c. &c. 

are the coordinates of /tt, / , /"/ , &c. when measured from the center of 
gravity, and it has been shown already that 

d 2 (x x) _ d^x 
~~d t 2 ~ dT 2 

d 2 (y-y) _ d^y 
dt 2 "dt s 

d 2 (z z) _ d 2 z 



dt ~dt 2 

and so on for the other bodies. Hence then it appears, that the motions 
of the bodies about the center of gravity, are the same as if there were but 
one force, scil. 

(//, + (i! + &c.) X distance 
and as if this force were placed in the center of gravity. 

Hence the bodies will all describe ellipses about the center of gravity, 
as a center; and their periodic times will all be the same. But their 
magnitudes, excentricities, the positions of the planes of their orbits, and 
of the major axes, may be of all varieties. 

Moreover the motion of any one body relative to any other, will be 
governed by the same laws as the motion of a body relative to a center 

of force, which force varies directly as the distance ; for if we take the 
equations 

- = (0, + ^ + &c.) (x x) 



12 A COMMENTARY ON [SECT. XL 

and subtract them we get. 

f|2 ( X v>^ 

( dt2 ^ = (A* + ft + &C.) (X - X ) 

and similarly 

d 2 (v v ) 

ch^ = ^ + /v + &c - } (- v - y ) 

and 

^s/ 7 _/\ 

^ t . = (* + ^ + &C.) (Z 2 ). 

Hence by composition and the general expression for force (yr|) ^ 

readily appears that the motion of & about p y is such as was asserted. 

470. Thus far relates merely to the motions of two bodies ; and these 
can be accurately determined. But the operations of Nature are on a 
grander scale, and she presents us with Systems composed of Three, and 
even more bodies, mutually attracting each other. In these cases the 
equations of motion cannot be integrated by any methods hitherto dis 
covered, and we must therefore have recourse to methods of approxi 
mation. 

In this portion of our labours we shall endeavour to lay before the 
reader such an exposition of the Lunar, Planetary and Cometary Theories, 
as may afford him, a complete succedaneum to the discoveries of our 
author. 

471. Since relative motions are such only as can be observed, we refer 
the motions of the Planets and Comets, to the center of the sun, and the 
motions of the Satellites to the center of their planets. Thus to compare 
theory with observations, 

// is required to determine the relative motion of a system of bodies, about 
a body considered as the center of their motions. 

Let M be this last body, /*, (* , /,", &c. being the other bodies of which 
is required the relative motion about M. Also let 

C, n, 7 
be the rectangular coordinates of M ; 

+ x, n + y, 7 + z; 
+x n + y ,7+z ; 

&c. 

those of ft, //, &c. Then it is evident that 

x, y, z ; 

Tff v / y 

x > y > z 

&c. 



BOOK I.] NEWTON S PRINCIPIA. 13 

will be the coordinates off*, & , &c. referred to M. 

Call ft /, & c . 

the distances of p, ///, &c. from M; then we have 



f = v (x * + y - 4- z i ) 

ft /> &c. being the diagonals of rectangular parallelepipeds, whose sides 
are 

x, y, z 

V f I/ 1> 

x j y > z 

&c. 
Now the actions of /w, / , ^", &c. upon M are 

At (jf [jf 

~~Z ) t 2 J "V7~2 ) O^^* 

and these resolved parallel to the axis of x, are 

V* X ft x. /A" x v 

7 F T r 7 7ir &c 

Therefore to determine , we have 

dT 2 ^ = 7 3 + "73" + T 7 ^ + &c. 



the symbol 2 denoting the sum of such expressions. 
In like manner to determine n, 7 we have 



dt 2 "7^ 

Q y $& z 

dl 2111 2 *7 T> 

The action of M upon /*, resolved parallel to the axis of x, and in the 
contrary direction, is 

_Mx 

Also the actions of ^ , A*", &c. upon ^ resolved parallel to the axis of x 
are, in like manner, 

tf (* x) ^ (*" . x ) ^" ( X /// _ x ) 

fd.m generally denoting the distance between ///" " and ///" 
But 



x y) 2 + ( 2 _ z ) 



to = V (X" X) 2 + (y"__ 
&C. = &C. 



A COMMENTARY ON [SECT. XL 



f 18 = V (X" x ) 2 + (y" y ) 2 + (z" 
and so on. 

Hence if we assume 

^ _ p.p. i^y." &c 

0,1 go,2 






i, 2 fl,3 



2,3 
&C. 

and taking the Partial Difference upon the supposition that x is the only 
variable, we have 

J- . (**\ = "x(x -) + (*> -x) &c> 



the parenthesis ( ) denoting the Partial Difference. Hence the sum of 
all the actions of / , /", &c. on /i is 

JL.fJil 

A* Vdx/ 
Hence then the whole action upon /A parallel to x is 

d. 2 (I + x) _ J_ xd_Xv MX 
d t 2 = /t6 Vdx/ ~ f 3 ; 
But 



d 2 x 1 /d >.\ MX ^x . 

" ~~" ^ 



Similarly, we have 



d t 2 " n Vdy^ ^ 3 g r 

1 /d Xx M z ^ z 

~^l 2 -TF (3) 



t 

If we change successively in the equations (1), (2), (3) the quantities 
,, x, y, z into 



(* , x , y , z ; 



..// v // v // // . 

f 5 x j y 5 z 

&c. 

and reciprocally ; we shall have all the equations of motion of the bodies 
^ /a", &c. round M. 



BOOK I.] NEWTON S PRINCIP1A. 15 

If we multiply the equations involving by M + 2. p ; that in x, by 
u. ; that in x , by /a , and so on ; and add them together, we shall have 

. d 2 ? /d x\ /dx\ /dXx d 2 x 

(M + s.^jpM ( dx ) + ( d ~, 

But since 

d Xx , g (X- - X) 



/ x = , g X - X & ^ 

\dx/ J 



. &c 



and so on in pairs, it will easily appear that 



x d 2 ^ d 2 x 

( M + 2 -^dT = - 2 ^dT^ 

whence by integrating we get 



d ? - c ^ d t - --Jl 

; ~ M + 2.^ M + 



and again integrating 

, 2. fj. x 

= a + ~ 



a and b being arbitrary constants. 
Similarly, it is found that 



These three equations, therefore, give the absolute motion of M in 
space, when the relative motions around it of p, /, /a.", &c. are known. 
Again, if we multiply the equations in x and y by 



and 

2 . ^ X- 



in like manner the equations in x and y by 



16 A COMMENTARY ON [SECT. XL 

and 

. 2. fjj X 



and so on. 

And if we add all these results together, observing that from the nature 
of X, (which is easily shown) 



and that (as we already know) 

/d X\ /d X\ 

2. (-y-) = 0, 2. (--) = 0, 
\dx/ \d y/ 

we have 

xd 2 y y d 2 x 2.,/*x d 2 y 

- -- - - 



y d 2 x 
" (J " 



M + 2,<T *dt 2 

and integrating, since 

/(xd 2 y yd 2 x) =/xd 2 y /yd 2 x 
= x d y yd xdy (ydx yd xdy) 
xdy ydx, 
we have 

x d v v d x 2,/u-x dy 

2 i^ const, -f- *r? 2 . (& . ** 

dt M + 2 . , dt 

2. [*> y d x 

~ M + 2. A& * 5Tt 

Hence 

,., xdy ydx xdy ydx dx 

C = M . 2 . IL . y . * U2 . A* X 2 u, . ^ 5-^ 1- 2 . A* y X 2 . /* -,- 



2 . ^ x X 2 . 



c being an arbitrary constant. 

In the same manner we arrive at these two integrals, 



(yy - y) (d 



7 and c" being two other arbitrary constants. 



BOOK I.] NEWTON S PRINCIPIA. 17 

Again, it we multiply the equation in x by 

i 
2 p d x 2 



2. fjt, d x 



**:-- - 
M + 2.^ 

the equation in y by 



-. 

M + 2.^ 
the equation in z by 

o j 2 . , d Z 

2 /* d z 2 p . .- - 
M + 2 . (J, 

it in like manner we multiply the equations in x , /, z by 



M + 2.^ 

2t 1 / r* t " A^ Cl V 

/// d y 2 u, . -_l-:_z *- 
M + 2. (j. 

/!/ ^v/ P /* d Z 

2 & d z 2 ,<// . =i-= : 

M + 2. (i, 9 

respectively, and so on tor the rest ; and add the several results, observ 
ing that 



we get 
2 v 



_ 
dt 2 = M + 2^" dt 



. 2 s . ^ d y ^ ^d 2 y 2 2 . ^ d z ^ >d 2 z 
2 h " 



and integrating, we have 

2 P T-T5 = const. + - p-jt- I 

d t 2 (M -f s/tt) d t 

+ 2 M 2 + 2 X, 

which gives 

, , dx +dy^dz* , . ( (dx r -dx) 2 +(d> 

- 5Tr - -+2.^.|V- ^_, , j 

f 2 M 2. -^- + 2x| (M + 2 /(*) 

i f J 

h being an arbitrary constant. 

VOL. IT. B 



18 A COMMENTARY ON [SECT. XI. 

These integrals being the only ones attainable by the present state of 
analysis, we are obliged to have recourse to Methods of Approximation, 
and for this object to take advantage of the facilities afforded us by the 
constitution of the system of the World. One of the principal of these 
is due to the fact, that the Solar System is composed of Partial Systems, 
formed by the Planets and their Satellites : which systems are such, that 
the distances of the Satellites from their Planet, are small in comparison 
with the distance of the Planet from the Sun : whence it results, that the 
action of the Sun being nearly the same upon the Planet as upon its Satel 
lites, these latter move nearly the same as if they obeyed no other action 
than that of the Planet. Hence we have this remarkable property, 
namely, 

472. The motion of the Center of Gravity of a Planet and its Satellites, 
is very nearly the same as if all the bodies formed one in that Center. 

Let the mutual distances of the bodies ^, & , p", &c. be very small 
compared with that of their center of gravity from the body M. Let 
also 

X = x + X, ; y = y -f- y, ; z = z + z,. 

x = x" + x/ ; y = y + y/; z = "z + z/; 

&c. 

x, y, z being the coordinates of the center of gravity of the system of 
bodies p, (if 9 ,", &c. ; the origin of these and of the coordinates x, y, z ; 
x , y , z , &c. being at the center of M. It is evident that x,, y /5 z, ; 
x/, y/, z/, &c. are the coordinates of (i>, p f , &c. relatively to their center of 
gravity ; we will suppose these, compared with x, y, z, as small quanti 
ties of the first order. This being done, we shall have, as we know by 
Mechanics, the force which sollicits the center of gravity of the system paral 
lel to any straight line, by taking the sum of the forces which act upon the 
bodies parallel to the given straight line, multiplied respectively by their 
masses, and by dividing this sum by the sum of the masses. We also 
know (by Mech.) that the mutual action of the bodies upon one another, 
does not alter the motion of the center of gravity of the system ; nor does 
their mutual attraction. It is sufficient, therefore, in estimating the forces 
which animate the center of gravity of a system, merely to regard the 
action of the body M which forms no part of the system. 

The action of M upon //., resolved parallel to the axis of x is 
MX 



BOOK I.] NEWTON S PRINCIPIA. 19 

the whole force which sollicits the center of gravity parallel to this straight 
line is, therefore, 



Substituting for x and g their values 

x It + x, 



U x + x /) 2 + (y + y/)H- ( z + Z /) 2 P 

If we neglect small quantities of the second order, scil. the squares and 
products of 

"V V 7 Y V <7 %7f* 

/ y / / / 5 y/ j "i > otc. 
and put 

7 = V (x 2 + P + z" 2 ) 

the distance of the center of gravity from M, we have 
- = * 4- 3- 3 x (x x, + "y y/ + z z,) 

e 3 f 3 ~s 3 7 3 

for omitting x 2 , y 2 &c., we have 

p = (i + X/ ) X K?) 2 + 2 (x x, + y y/ + z Z/ )} ~ f nearly 

= (x+x y ) X J(7) ~ 3 3 (7) ~ 5 ( x x / + y y/ + z~zj nearly 

x + x/ 3 x - - 

= "^\T~ "" ( x x / + y y/ "i z z /) nearly. 

Again, marking successively the letters x /5 y /s z /5 with one, two, three, 
&c. dashes or accents, we shall have the values of 



But from the nature of the center of gravity 



, 

__ _i nearJv 
3 



Thus the center of gravity of the system is sollicited parallel to the 
axis of x, by the action of the body M, very nearly as if all the bodies of 
the system were collected into one at the center. The same result evi 
dently takes place relatively to the axes of y and z; so that the forces, by 

B2 



20 A COMMENTARY ON [SECT. XL 

which the center of gravity of the system is animated parallel to these 
axes, by the action of M, are respectively 

My Mz 

"6) ;a "& 

When we consider the relative motion of the center of gravity of the 
system about M, the direction of the force which sollicits M must be 
changed. This force resulting from the action of (*, p, &c. upon M, and 
resolved parallel to x, in the contrary direction from the origin, is 



if we neglect small quantities of the second order, this function becomes, 
after what has been shown, equal to 

X 2./C6 

I 3 

In like manner, the forces by which M is actuated arising from the 
system, parallel to the axes of y, and of z, in the contrary direction, are 



It is thus perceptible, that the action of the system upon the body M, 
is very nearly the same as if all the bodies were collected at their common 
center of gravity. Transferring to this center, and with a contrary sign, 
the three preceding forces; this point will be solicited parallel to the 
axes of x, y and z, in its relative motion about M, by the three following 
forces, scil. 



- (M + ?(*) _(M + 2^) -~ y _ (M + 2. 

(sr (s) 3 (s) 3 

These forces are the same as if all the bodies /, ft , /*", &c. were col 
lected at their common center of gravity; which center, therefore, moves 
nearly (to small quantities of the second order] as if all the bodies were col 
lected at that center. 

Hence it follows, that if there are many systems, whose centers of gra 
vity are very distant from each other, relatively to the respective distances 
of the bodies of each system ; these centers will be moved very nearly, as 
if the bodies of each system were there collected ; for the action of the 
first system upon each body of the second system, is the same very nearly 
as if the bodies of the first system were collected at their common center 
of gravity ; the action of the first system upon the center of gravity of the 
second, will be therefore, by what has preceded, the same as on this hy 
pothesis ; whence we may conclude generally that the reciprocal action of 



BOOK L] NEWTON S PRINCIPIA. 21 

different systems upon their respective centers of gravity > is the same as if all 
the bodies of each system were there collected, and also that these centers 
move as on that supposition. 

It is clear that this result subsists equally, whether the bodies of eacli 
system be free, or connected together in any way whatever ; for their mu 
tual action has no influence upon the motion of their common center 
of gravity. 

The system of a planet acts, therefore, upon the other bodies of the 
Solar system, very nearly the same as if the Planet and its Satellites, 
were collected at their common center of gravity; and this center itself is 
attracted by the different bodies of the Solar system, as it would be on 
that hypothesis. 

Having given the equations of motion of a system of bodies submitted 
to their mutual attraction, it remains to integrate them by successive 
approximations. In the solar system, the celestial bodies move nearly as 
if they obeyed only the principal force which actuates them, and the per 
turbing forces are inconsiderable; we may, therefore, in a first approxi 
mation consider only the mutual action of two bodies, scil. that of a planet 
or of a comet and of the sun, in the theory of planets and comets ; and 
the mutual action of a satellite and of its planet, in the theory of satellites. 
We shall begin by giving a rigorous determination of the motion of two 
attracting bodies : this first approximation will conduct us to a second in 
which we shall include the first powers of small quantities or the perturb 
ing forces ; next we shall consider the squares and products of these 
forces; and continuing the process, we shall determine the motions of the 
heavenly bodies with all the accuracy that observations will admit of. 



FIRST APPROXIMATION. 



478. We know already that a body attracted towards a fixed point, 
by a force varying reciprocally as the square of the distance, de 
scribes a conic section ; or in the relative motion of the body p, round 
M, this latter body being considered as fixed, we must transfer in a di- 
rection contrary to that of p, the action of/* upon M; so that in this re 
lative motion, p is solicited towards M, by a force equal to the sum ol " 
the masses M, and i* divided by the square of their distance. All this 
has been ascertained already. But the importance of the subject in the 
Theory of the system of the world, will be a sufficient excuse for repre 
senting it under another form, 

B3 



22 A COMMENTARY ON [SECT. XL 

First transform the variables x, y, z into others more commodious for 
astronomical purposes, g being the distance of the centers of p and M, 
call (v) the angle which the projection of g upon the plane of x, y makes 
with the axis of x; and (6) the inclination of g to the same plane; we 
shall have 

x = f cos. 6 cos. v ; -\ 

y = g cos. 6 sin. v; V ........ (1) 

z = g sin. 6. } 



Next putting 

we have 
/dQ 


M + , 


* o 


ex + yy + zz 7 ) 


e 

!\ 1 /d Xx 


M+/ 

M 


f /3 


dx 


1/1^ 


2 . 7 3 
^X 



Similarly 






Q\ _ 1 /dj^x M 
~ 



^x _ _ v ftz 
/ ~ g 3 g 3 
Hence equations (1), (2), (3) of number 471, become 

d 2 x /dQx d^y /d_Qx d z _ /dQv 
dt 2 \dx-/ ; dt 1 ~ : \dy) d t 2 ~ VdzJ 

Now multiplying the first of these equations by cos. 6. cos. v; the 
second by cos. 6. sin. v ; the third by sin. 6, we get, by adding them 



In like manner, multiplying the first of the above equations by g cos.0 X 
sin. v; the second by g cos. 6 cos. v and adding them, &c. we have 



inr 

And lastly multiplying the first by g sin. 6. cos. v ; the second by 
g sin. 6. cos. v and adding them to the third multiplied by cos. 6. we 
have 



To render the equations (2), (3), (4), still better adapted for use, let 

1 

u ~ 

g cos. 



BOOK I.] 



NEWTON S PRINCIPIA. 



23 



and 

s = tan. 6 

u being unity divided by the projection of the radius g upon the plane 
of x, y ; and s the tangent of the latitude of (A from that same plane. 
If we multiply equation (3) by g z d v cos. 2 6 and integrate, we get 



h being the arbitrary constant. 
Hence 

d t = 






d 



j 
d 



(5) 
^ 



\ u 



If we add equation (2) multiplied by cos. 6 to equation (4) multi 
plied by - , we shall have 



di 

u 



1 d 



whence 

r d u \ d v . , f /d Q\ s /d Q\ 1 

.^n)+^r, = " Qdt {(^) + 7r(ds)}- 

Substituting for d t, its foregoing value, and making d v constant, we 
shall have 

o77 nr .... (6) 



= 



d v 



d v / u 2 
In the same way making d v constant, equation (4) will become 

dQ 



= 



() _/) 

d * s- . . d v \ d v / \ d u/ v ; 



d s 



d v 



. . . (7) 



Now making M + ^ = m, we have (in this case) 

f-^ m m u 

Q = or = r-r- r 

g V (1 + s 2 ) 

and the equations (5), (6), (7) will become 
dv 



dt = 



h.u 



0=^ + u_ 



h s (l 



= 



(8) 



24 A COMMENTARY ON [SECT. XI. 

(These equations may be more simply deduced directly 124 and Wood- 
house s Phys. Astron.) 

The area described during the element of time d t, by the projection 
of the radius-vector is i-? ; the first of equations (8) show that tins area 

is proporti&nal to that element, and also that in a finite time it is propor 
tional to the time. 

Moreover integrating the last of them (by 122) or by multiplying by 
2 d s, we get 

s = y sin. (v 0) ......... (9) 

7 and 6 being two arbitrary constants. 

Finally, the second equation gives by integration 

U = h*(l+V) ^ 1 + S " + ecos.(v- w )} = V1 + s "; . . . (10) 
e and -a being two new arbitraries. 

Substituting for s in this expression, its value in terms of v, and then 
this expression in the equation 



the integral of this equation will give t in terms of v ; thus we shall have 
v, u and s in functions of the time. 

This process may be considerably simplified, by observing that the 
value of s indicates the orbit to lie wholly in one plane, the tangent of 
whose inclination to a fixed plane is 7, the longitude of the node 6 bein^ 
reckoned from the origin of the angle v. In referring, therefore, to this 
plane the motion of//,; we shall have 

s = and 7=0, 
which give 

] p 

u = ? = pU + ecos - (v OJ. 

This equation is that of an ellipse in which the origin of g is at the 
focus : 



is the semi-axis-major which we shall designate by a; e is the ratio of 
the excentricity to the semi-axis-major ; and lastly * is the longitude of 
the perihelion. The equation 

d v 



d t = _ 
h u 



BOOK L] NEWTON S PRINCIPIA. 25 

hence becomes 

d - a^(l e 2 )^ d v 

V^fJi, {1+ecos. (v &)}" 

Develope the second member of this equation, in a series of the angle 
v -a and of its multiples. For that purpose, we will commence by 
developing 

1 

1 + e cos. (v w) 
in a similar series. If we make 



X = 



1 + V ( 1 e 2 ) 
we shall have 

1 1_ _f 1 X. c -(v-*Q^- 

1+ecos. (v w)~ y l e* I l + Xc( v ~ w ) 1 1 + Xc ( v ^OV 

e being the number whose hyperbolic is unity. Developing the second 
member of this equation, in a series; namely the first term relatively 
to powers of c( v ~ w ) v/ i 1 , and the second term relatively to powers of 
c ~ ( v **") y l and then substituting, instead of imaginary exponentials, 
their expressions in terms of sine and cosine ; we shall find 



I + e cos. (v -af] " V 1 e 2 

{I 2 X cos. (v w ) + 2 X 2 cos. 2 (v ) 2 X 3 cos. 3 (v *) + &c.| 
Calling <p the second member of this equation, and making q = ; we 
shall have generally 

i = e " " ldm (T 

? 1 + e coa, (v ~)} m + L 1.2.3 in. d q M 

for putting 



q q + R 
R being = cos. (v w) 

*-e) 



1 



(q + R) 2 

** (j) _^ 
dq 2 -(q + il) 
&c. = &c. 



26 A COMMENTARY ON [SECT. XI. 



___ 4- 2 - 3 ____ m 

~~~ 



(q 



_ 

dq ra 2.3...m~ (q+ R) m + 

1 



1 + e cos. (v w)J m + 1 " 
Hence it is easy to conclude that if we make 

U + e cos. (v w )f = (* e 2 ) 

[I + E ). cos. (v ,) + E (2 lcos. 2 (v ~) + &c.| 
we shall have generally whatever be the number (i) 



(1 + V 1 e 2 ) 1 

the signs + being used according as i is even or odd ; supposing there 
fore that u r= a~ a V m, we have 

ndt = dv [I + E (1 >cos. (v ) + E (2 > cos. 2 (v )+ &c.* 
and integrating 

n t +e = v + E (1) sin. (v ) + \ E (2) sin. 2 (v ) + &c. 
e being an arbitrary constant. This expression for n t + is very con 
vergent when the orbits are of small excentricity, such as are those of the 
Planets and of the Satellites ; and by the Reversion of Series we can find 
v in terms of t : we shall proceed to this presently. 

474. When the Planet comes again to the same point of its orbit, v is 
augmented by the circumference 2 it ; naming therefore T the time of the 
whole revolution, we have (see also 159) 

T-i- -^. 
n V m 

This could be obtained immediately from the expression 

T 1 J 

~TT~ 

2 area of Ellipse _ 2jra b 
~~h~ IT 

But by 157 

h s = m a ( 1 e 2 ) 

2_ 2 
_ it a 

V m 



BOOK I.] 



NEWTON S PRINCIPIA. 



27 



If we neglect the masses of the planets relatively to that of the sun we 
have 



which will be the same for all the planets j T is therefore proportional in 

that hypothesis to a 2 , and consequently the squares of the Periods are as 
the cubes of the major axes of the orbits. We see also that the 
same law holds with regard to the motion of the satellites around their 
planet, provided their masses are also deemed inconsiderable compared 
with that of the planet. 

475. The equations of motion of the two bodies M and fj<> may also be 
integrated in this manner. 

Resuming the equations (1), (2), (3), of 471, and putting M + /* = m, we 
have for these two bodies 



= 



~ 



= 



dt 2 
!!_? 

dt 2 
d 2 z 

dt 2 



x m x 

- 3 



m y 



m z 



(0) 



The integrals of these equations will give in functions of the time t, the 
three coordinates x, y, z of the body & referred to the center of M ; we 
shall then have (471) the coordinates , n, 7 of the body M, referred to a 
fixed point by means of the equations 



- a 



bt 



* x 
m 



H = a 7 + b 7 t 



= a " + b"t 



m 



Lastly, we shall have the coordinates of ^ referred to the same fixed 
point, by adding x to , y to n, and z to y : We shall also have the rela 
tive motion of the bodies M and /*, and their absolute motion in space. 

476. To integrate the equations (0) we shall observe that if amongst 

the (n) variables x^, x ^ 2) x (n) and the variable t, whose difference 

is supposed constant, a number n of equations of the following form 



= f 



H . X 



dt dt 1 - dt- 

in which we suppose s successively equal to 1, 2, 3 n ; A, B H 

oeing functions of the variables x (1) , x (2) , &c. and of t symmetrical 



28 



A COMMENTARY ON 



[SECT. XL 



with regard to the variables x (1) , x (2 >, &c. that is to say, such that they 
remain the same, when we change any one of these variables to any other 
and reciprocally ; suppose 

x (1) __ a (I) x (n - i + 1) _j_ b (1) x (x - i + 2) _f_ h (1) x (n) ? 

X (2) = a (2) x (n-i + l) _|_ b (2) x (n-i + 2) _^. h (2 ) X n . 



x a X ~T~" x ~T" ft x. 

a (1) , b (n , h (1) ; a (2) , b (2) , &c. being the arbitraries of which the 

number is i (n i). It is clear that these Values satisfy the proposed 
system of equations : Moreover these equations are thereby reduced to i 

equations involving the i variables x (n ~ i + 1) x w . Their integrals 

will introduce i 2 new arbitraries, which together with the i (n i) pre 
ceding ones will form i n arbitraries which ought to give the integration 
of the equations proposed. 

477. To apply the above Theorem to equations (0) ; we have 

z = a x + b y 

a and b being two arbitrary constants, this equation being that of a plane 
passing through the origin of coordinates ; also the orbit of ^ is wholly in 
one plane. 

The equations (0) give 



(0 ) 




g x 2 + y 2 + z 



Also since 
and 
and differentiating twice more, we have 



and consequently 



+ 3(dxd 2 x + dyd 2 y + dzd 2 z), 
d 3 x d 3 y d 3 

d 2 x . d 2 v . 



Substituting in the second member of this equation for d 3 x, d 3 y, d z 



BOOK L] NEWTON S PRJNCIPIA. 29 

their values given by equations ((X), and for d 2 x, d 2 y, d 2 z their values 
given by equations (0) ; we shall find 



If we compare this equation with equations (0 ), we shall have in virtue 
of the preceding Theorem, by considering -y , -^ , -.- , y|, as so many 

particular variables x (1) , x (2 \ x (3) , x W, and g as a function of the time t; 

d g = A d x + y d y ; 
X and 7 being constants ; and integrating 

h 2 
= -. + Xx + 7 y, 

h 2 

being a constant. This equation combined with 

z = ax + by;g 2 = x 2 + y 2 + z* 

gives an equation of the second degree in terms of x, y, or in terms of 
x, z, or of y, z; whence it follows that the three projections of the curve 
described by p about M, are lines of the second order, and therefore that 
the curve itself (lying in one plane) is a line of the second order or a conic 
section. It is easy to perceive from the nature of conic sections that, the 
radius-vector g being expressed by a linear function of x, y, the origin of 
x, y ought to be in the focus. But the equation 

h 2 

e = m + Xx + y y 

gives by means of equations (0) 

( V\ 

_ d 2 g , V mJ 

= d^ + ^- p- 

Multiplying this by d g and integrating we get 



a being an arbitrary constant. Hence 
dt= 



I / e \ 

m J (2 g -r ) 

V v a m / 

which will give g in terms of t ; and since x, y, z are given above in terms 
of g, we shall have the coordinates of ^ in functions of the times. 

478. We can obtain these results by the following method, which has 
the advantage of giving the arbitrary constants in terms of the coordinates 
x, y, z and of their first differences ; which will presently be of great use 
to us. 



30 A COMMENTARY ON [SECT. XI. 

Let V = constant, be an integral of the first order of equations (0), V 
being a function of X, y, z, -, - , -r^ , i-? . Call the three last quantities 

x , y , z . Then V = constant will give, by taking the differential, 

/d Vx dx ,d V, d y ,dVx d z 
VdxV dt " VdyJ dt " Vdz dt 

r ( LY\ dx/ j f d J^ d y f d v d z/ 

" Vd xV dt " " Vd yV dt" " " \d z ) "dt" 
But equations (0) give 

d x m x d y m y d z m z 

dT : ~7 r "dT : ~7 dT : "~p~ ; 

we have therefore the equation of Partial Differences 



, 

= x 



, /d Vx 

(a) 



m /dV 



It is evident that every function of x, y, z, x r , y . z which, when sub 
stituted for V in this equation, satisfies it, becomes, by putting it equal to 
an arbitrary constant, an integral of the first order of the equations (0). 

Suppose 

V = U + U + U" + &c. 

U being a function of x, y, z; U a function of x, y, z, x , y , z but of the 
first order relatively to x , y , z ; U" a function of x, y, z, x x , y , z and of 
the second order relatively to x , y , z , and so on. Substitute this value 
of V in the equation (I) and compare separately 1. the terms without 
x , y , z ; 2. those which contain their first powers ; 3. those involving their 
squares and products, and so on ; and we shall have 

U x /d U x /d U 



m f /d U"x /d U 

= x 



, /d U x , ,/d U\ , 7 /d U x m / /d U w x w /dU 

x "- 



m 



&c. 
which four equations call (F). 

The integral of the first of them is 

U = ftmct. Jx y y x , x z z x , y z z y , x, y, z] 



BOOK I.] NEWTON S PRINCIPIA. 31 

The value of U 7 is linear with regard to x 7 , y 7 , z 7 ; suppose it of this 
form 

U = A (x y y x 7 ) + B (x z 7 z x ) + C (y z z y 7 ) ; 
A, B, C being arbitrary constants. Make 

U" 7 , &c. = 0; 
then the third of the equations (F) will become 



The preceding value of (J 7 satisfies also this equation. 
Again, the fourth of the equations (F) becomes 



of which the integral is 

U 7 = funct. x y 7 y x 7 , x z 7 z x 7 , y z 7 z y 7 , x 7 , y 7 , z 7 }. 
This function ought to satisfy the second of equations (F), and the first 
member of this equation multiplied by d t is evidently equal to d U. The 
second member ought therefore to be an exact differential of a function of 
x, y, z ; and it is easy to perceive that we shall satisfy at once this condi 
tion, the nature of the function U 77 , and the supposition that this function 
ought to be of the second order, by making 
U 77 = (D y 7 E x ) . (x y 7 y x 7 ) + (D z 7 F x 7 ) (x z 7 z x 7 ) 

+ (E z F y ) (y z z y ) + G (x 2 + y 2 + z 2 )j 

D, E, F, G being arbitrary constants ; and then g being = V 7 x 2 +y 2 +z 2 , 
we have 

U = -- m (Dx + Ey+ Fz + 2G); 

Thus we have the values of 

U, U , U" ; 
and the equation V = constant will become 

const. = m D x+E y+F z + 2 G} + (A + D y E x 7 ) (x y y X ) 

+ (B + D z F x ) (x z z x 7 ) + (C + E z 7 F y) (y z z y) 

+ G (x 2 + y 2 + z 2 ). 

This equation satisfies equation (I) and consequently the equations (0) 
whatever may be the arbitrary constants A, B, C, D, E, F, G. Sup 
posing all these = 0, 1. except A, 2. except B, 3. except C, &c. and 
putting 

d x d y d z 

d! dt t ,l for * ,y,z, 



A COMMENTARY ON [SECT. XI. 



we shall have the integrals 



(P) 



c 


d y > d x fl x 


d 


z 


zdx 


, C /X 

dy 




j d z 


zd 


y 




dt 


, V, 

d y 2 ~K 


d 


dt 
z 2 ) 


1 y 


.d 


x 


d t 
z d z . 


dx 


= f 

f" 


\ s 
-1 yf m 


dx 


2 + 


d 


z 2 ) 


1 x 


d t 
dx 


.d 


y . 


z d z. 


dy 


1 } If 

, r m 


dx 


dt 2 
2 + 


d 


J 

y^ 


T 

1 x 


dt 
d x 


2 
.d 


i 

z | 


dt 

y dy. 


i 

dz 


n- 


j- x 


d + 




1 


T^ 


2 dl 




T^ 


dt 





c, c , c", f, f , f" and a being arbitrary constants. 

The equations (0) can have but six distinct integrals of the first order, 
by means of which, if we eliminate d x, d y, d z, we shall have the three 
variables x, y, z in functions of the time t; we must therefore have at least 
one of the seven integrals (P) contained in the six others. We also per 
ceive d priori, that two of these integrals ought to enter into the five 
others. In fact, since it is the element only of the time which enters 
these integrals, they cannot give the variables x, y, z in functions of the 
time, and therefore are insufficient to determine completely the motion of 
about M. Let us examine how it is that these integrals make but five 
distinct integrals. 

Z Q V ^ __ V d Z 

If we multiply the fourth of the equations (P) by -- *^ * - , and 

Y H z ___ Z Cl X 

add the product to the fifth multiplied by - j - , we shall have 



n f z dy ydz f , xdz zdx xdy y dx f m d x 2 -f d y 2 ) 

~~dT~ ~dT~ "Tt~ IT" dt 2 j 

xdy ydxfxdx. dz 

~d~r ~ i ~~d T 2 ~ 



d.dz 



yy 

d 

xdy ydx xdz zdx ydz zdy , . 

Substituting for - d t - - g~j: - > - 3~ - - their 

values given by the three first of the equations (P), we shall have 

f c f c" ( m d x 2 + d yg \ x d x . d z y d y.d z 
~^~ Z \7" "dT^" J dt e ~~d"t^~ 

This equation enters into the sixth of the integrals P, by making 

f" = f/ c/ ~ f c " or = f c" f c + f" c. Also the sixth of these 
c 

integrals results from the five first, and the six arbitraries c, c , c 7 , f, f, f" 
are connected by the preceding equation. 



BOOK I.] NEWTON S PRINCIPIA. 33 

If we take the squares off, P, f" given by the equations (P), then add 
them together, and make f 2 + P 2 -f- P 2 = 1 2 , we shall have 
-/ * dx 2 +dy 2 +dz 2 



+ dy 2 +dz g m i 
d~t 2 ~J 



dt s Vdt/ J * \ d~ 

but if we square the values of c, c , c", given by the same equations, and 
make c 2 + c /2 + c" 2 = h 2 ; we get 



dt 2 
the equation above thus becomes 

d x 2 + dy 2 + d z 2 2m m 2 I 2 
dt 2 "7" " h 2 * 

Comparing this equation with the last of equations (P), we shall have 
the equation of condition, 

m 2 I 2 _m 

h* a * 

The last of equations (P) consequently enters the six first, which are 
themselves equivalent only to five distinct integrals, the seven arbitrary 
constants, c, c , c", f, P, f", and a being connected by the two preceding 
equations of condition. Whence it results that we shall have the most 
general expression of V, which will satisfy equation (I) by taking for this 
expression an arbitrary function of the values of c, c , c", f, and P, given 
by the five first of the equations (P). 

479. Although these integrals are insufficient for the determination of 
x, y, z in functions of the time ; yet they determine the nature of the 
curve described by ft about M. In fact, if we multiply the first of the 
equations (P) by z, the second by y, and the third by x, and add the 
results, we shall have 

= c z c y -f- c" x, 
the equation to a plane whose position depends upon the constants 

c c c" 
c, c , c . 

If we multiply the fourth of the equations (P) by x, the fifth by y, and 
the sixth by z, we shall have 



but by the preceding number 

, dx 2 + dy*+ dz 2 






dt 2 dt 2 

.-. = m g h 2 + f x + f 7 y + f" z. 
This equation combined with 

= c" x c y + c z 
VOL. II. C 



34 A COMMENTARY ON [SECT. XI. 

and 

g = x 2 + y 2 + z 2 

gives the equation to conic sections, the origin of being at the focus. 
The planets and comets describe therefore round the sun very nearly 
conic sections, the sun being in one of the foci ; and these stars so move 
that their radius-vectors describe areas proportional to the times. In fact, 
if d v denote the elemental angle included by , g + d f, we have 

d x 8 + d y 2 + d z 2 = s z d v 2 + d s 2 
and the equation 



dt 2 dt 2 

becomes 

* 4 d v 2 = h 2 d t 1 ; 

hdt 
,.dv=_. 

Hence we see that the elemental area I 2 d v, described by f, is propor 
tional to the element of time d t ; and the area described in a finite time is 
therefore also proportional to that time. We see also that the angular 

motion of ^ about M, is at every point of the orbit, as - z ; and since without 

sensible error "we may take very short times for those indefinitely small, we 
shall have, by means of the above equation, the horary motions of the planets 
and comets, in the different points of their orbits. 

The elements of the section described by p, are the arbitrary constants 
of its motion ; these are functions of the arbitraries c, c , c", f, P, f", and 

. Let us determine these functions. 
a 

Let 6 be the angle which the intersection of the planes of the orbit and 
of (x, y) makes with the axis of x, this intersection being called the line 
of the nodes ; also let <p be the inclination of the planes. If x , y be the 
coordinates of //. referred to the line of the nodes as the axis of abscissas, 
then we have 

x = x cos. 6 + y sin. 6 
y = y cos. d x sin. 6. 
Moreover 

z = y tan. <p 

.: z = y cos. 6 tan. <p x sin. 6 tan. <p. 
Comparing this equation with the following one 
= c" x c y + c / 



BOOK L] NEWTON S PRINCIPIA. 35 

we shall have 

c = c cos. 6. tan. p 

c" = c sin. 6 tan. <p 
whence 

c" 

tan. d = 
c 

and 

t,n. ?= V(c"j L 

C 

Thus are determined the position of the nodes and the inclination of the 
orbit, in functions of the arbitrary constants c, c , c". 
At the perihelion, we have 

g d g = 0, orxdx + ydy + zdz 0. 

Let X, Y, Z be the coordinates of the planet at this point ; the fourth 
and the fifth of the equations (P) will give 
_Y _ P 

-A. I 

But if I be called the longitude of the projection of the perihelion upon 
the plane of x, y this longitude being reckoned from the axis of x, we have 

Y 

v- = tan. I ; 



which determines the position of the major axis of the conic section. 
If from the equation 

dx+ d y 2 + d Z * g*dg 
d t 2 d t 2 

,. . dx 2 + d y 2 + d z 2 

we eliminate - - p^ - , by means of the last of the equa 

tions (P), we shall have 



but d is at the extremities of the axis major ; we therefore have at these 
points 



v, 2 

- e*_ 2a S+ --- 
m 



The sum of the two values of g in this equation, is the axis major, and 
their difference is double the excentricity ; thus a is the semi-axis major of 
the orbit, or the mean distance of p from M ; and 



36 A COMMENTARY ON [SECT. XI. 

is the ratio of the excentricity to the semi-axis major. Let 

/y ^ ma/ 
and having by the above 

m _ m 2 1 2 
"a" "IT 2 ; 

we shall get 

m e = 1. 

Thus we know all the elements which determine the nature of the conic 
section and its position in space. 

480. The three finite equations found above between x, y, z and g give 
x, y, z in functions of g ; and to get these coordinates in functions of the 
time it is sufficient to obtain g in a similar function ; which will require a 
new integration. For that purpose take the equation 



f , 

-- =_ h 

dt 2 



But we have above 



h 2 = -- (m 2 I 2 ) = am (1 e 2 ); 

P d P 
... d t = L-J _ . 

V m I 12 g S - a (1 e 2 ) V 

whose integral (237) is 

t + T = ~ (u e sin. u) (S) 

/I s \ 
u being = cos. 1 f ) an< ^ ^ an arbitrary constant. 

This equation gives u and therefore g in terms of t; and since x, y, z 
are given in functions of g, we shall have the values of the coordinates for 
any instants whatever. 

We have therefore completely integrated the equations (0) of 475, and 
thereby introduced the six arbitrary constants a, e, I, 6, <p, and T. The 
two first depend upon the nature of the orbit ; the three next depend upon 
its position in space, and the last relates to the position of the body u. 
at any given epoch ; or which amounts to the same, depends upon the 
instant of its passing the perihelion. 

Referring the coordinates of the body ^, to such as are more commodious 
for astronomical uses, and for that, naming v the angle which the radius- 



BOOK I.] NEWTON S PRINCIPIA. 37 

vector makes with the major axis setting out from the perihelion, the 
equation to the ellipse is 

a (1 e 2 ) 
1 + e cos, v 

The equation 

g = a ( 1 e cos. u) 

indicates that u is at the perihelion, so that this point is the origin of two 
angles u and v ; and it is easy hence to conclude that the angle u is formed by 
the axis major, and by the radius drawn from its center to the point where 
the circumference described upon the axis major as a diameter, is met by 
the ordinate passing through the body p at right angles to the axis major. 
Hence as in (237) we have 

v 1 1 + e _ u 

tan - 2 = ^T=-e taU -2 



We therefore have (making T = 0, &c.) 
n t = u e sin. u 
= a ( 1 e cos. u) 



and 



v / 1 + e u 



(0 



n t being the Mean Anomaly, 

n the Excentric Anomaly, 
v the True Anomaly. 

The first of these equations gives u in terms of t, and the two others 
will give g and v when u shall be determined. The equation between u 
and t is transcendental, and can only be resolved by approximation. 
Happily the circumstances attending the motions of the heavenly bodies 
present us with rapid approximations. In fact the orbits of the stars are 
either nearly circular or nearly parabolical, and in both cases, we can de 
termine u in terms of t by series very convergent, which we now proceed 
to develope. For this purpose we shall give some general Theorems 
upon the reduction of functions into series, which will be found very use 
ful hereafter. 

481. Let u be any function whatever of , which we propose to deve 
lope into a series proceeding by the powers of a. Representing this 
series by 

U = > a.q,+ a s .cj 8 + a". q n + a n + | . q D+ + &c. 

C3 



38 A COMMENTARY ON [SECT. XL 

"j qi> q-25 & c - being quantities independent of a, it is evident that u is what 
u will become when we suppose a = ; and that whatever n may be 

= 1.2....n.q n + 2.3....(n+l).a.q n + 1 + &c. 



/d n u\ 
the difference ( ~ -J being taken on the supposition that every thing in 

u varies with a. Hence if we suppose after the differentiations, that a = 0, 

, . /d n u\ 

in the expression (-, -J we have 

d n u\ 

X 



1.2 ____ n 

This is Maclaurin s Theorem (see 32) for one variable. 
Again, if u be a function of two quantities a, a , let it be put 
u = U + a . q 1)0 + a 2 . q 2 ; + &C. 
+ <* . qo,i + qi,i + &c. 

+ 2 - qo, 2 + &c. 
the general term being 



tt a n q n .n 



Then if generally 

/ d n + n u 

\d n . d u n 

denotes the (n + n ) th difference of u, the operation being performed (n) 
times, on the supposition that a is the only variable, and then n times on 
that of a! being the only variable, we have 

a 2 q3>0 "*" 4 a 3 q4>0 + 5 a * q5>0 + &c 

.a / q 2 ,i +3a 2 a q 3jl +4a 3 a / q 4>1 + &C. 
a /2 22 +3a s a / + &c. 



2 a ^. + 4< 3 a 2 ^.o + 5. 4 a 3 q 5>0 + &c. 

2 q 2> , + 3. 2aaq 3>1 + 4. 3a 2 aq 4jl + 8cc. 
+ 2 a 2 2 3.2aa 2 &C. 



T&) = 2 q 2)1 + 3. 2 a q 3)1 + &c. 

+ 2 a q 2>2 + &c. 
and continuing the process it will be found that 



Tjr = 2- 3- . . . n X 2. 3. . ..n X q, n , 



BOOK I.] NEWTON S PRINCIPIA. 39 

so that when , a both equal 0, we have 
/ d B + " u N 

Vd .d ". / 

q " n/ -27^77.. n x 2.3....n ^ 

And generally, if u be a function of a, d, a", &c. and in developing it 
into a series, if the coefficient of . & " . " n ". & c . be denoted by q n , n ,, ., &c 
we shall have, in making , d, a", &c. all equal 0, 

( c ] n + n + n" + &c. u 
_ d".dd" .da""",&C.) 

] ;" "> "" - 2.3....n X -273- . . . n X 2. 3 . . . . n" X &c. (2) 

This is Maclaurin s Theorem made general. 

482. Again let u be any function of t + , t + , t" + a", &c. and 
put 

u = g> (t + a, t + a, t" + a", &c.) 
then since t and a are similarly involved it is evident that 

d n + " + "" + &c - . u \ __ / d n + n + n " + &c - . u \ 
Vd a n . d a. n/ . d //n "~&c"./ = \d t n . d t /n . d t" n ". &c./ 
and making 

, A, a", &c. = 0, 



or 



u = <p (t, t 7 , t", &c.) 
by (2) of the preceding article we have 

, t , t", &c.)x 
d t n . d t /n< . d t" n " &c / 



. . 
ln n/>n " &c ~ 2.3.. ..n X 2. 3 . . . . n 

which gives Taylor s Theorem in all its generality (see 32). 
Hence when 

u = <f> . (t + ) 

d n .?(t) 

" 2.3 ____ n.dt" 
and we thence get 

( + .) = > (t) + .!^fil + "_ ! .^ + & c ...... (i) 

483. Generally, suppose that u is a function of , , a x , & c . and of 
tj t r , t", &c. Then, if by the nature of the function or by an equation of 
R. ftial Differences which represents it, we can obtain 

/ d n + n/ + &c - . u v 
Vda". da" . &cJ 

in a function of u, and of its Differences taken with regard to t, t , &c. 



40 A COMMENTARY ON [SECT. XI. 

calling it F when for u we put u or make a, a, a", &c. = ; it is evident 

we have 

_ F _ 

qn.n .n-.to. ~ g. 3 . . . n X g. 3 . . . n X 2. 3 . . . n", X &C. 

and therefore the law of the series into which u is developed. 

For instance, let u, instead of being given immediately in terms of a, 
and t, be a function of x, x itself being deducible from the equation of 
Partial Differences 



in which X is any function whatever of x. That is 
Given 

u = function (x) 
d 



to develop e u into a series ascending by the powers of a. 
First, since 



/dux P /d/Xd_Ux 

Vd a) ~ \ d t ) 
Hence 



_ 
a*)~ \ da.dt J 

But by equation (k), changing u into J X d u 
,d./Xdu x _ /d./X 2 dux 

v do )~\ dt ; 

. f d u \ _ /d 2 /X 2 dux 
V d aV ~ \ d t 8 / 
Again 

/d 3 ux __ /d 3 /X 2 dux 

\da 3 J-l da.dt 2 / 

But by equation k, and changing u into f X 2 d u 
/d/X*dux _ /d/X 3 dux 

\ d y~v dt J 

/d uv /d 3 ./^^_d_ux 

Vd^v-V dt 3 r 

Thus proceeding we easily conclude generally that 



Now, wlien a = 0, let 

x = function of t = T 



m 



BOOK I.] NEWTON S PRINCIPIA. 41 

and substitute this value of x in X and u ; and let these then become X 
and u respectively. Then we shall have 



. 

/cPMiN = 

\da n / d t"- 1 

and 



" A d T~ /0 . 

* q " -"2737 .ndt- 1 (2) 



which gives 



, . 

du , a 2 >> d t/ , a 

- d - t + T . dl + .. 



which is Lagrange s Theorem. 

To determine the value of x in terms of t and a, we must integrate 



In order to accomplish this object, we have 
and substituting 
we shall have 



d x = \d t + X d .; 



( x \ 

i ^d t / 
.-. d x = ^ 



which by integration, gives 

x = p (t + a X) . (2) 

<p denoting an arbitrary function. 

Hence whenever we have an equation reducible to this form x = 
f (t + X), the value of u will be given by the formula (p), in a series of 
the powers of a. 

By an extension of the process, the Theorem may be generalized to the 
case, when 

u = function (x, x , x", &c.) 



42 A COMMENTARY ON [SECT. XI. 

and 

x = p (t + a X) 
x = ? (t + X ) 
x" = p" (t" + a" X") 
&c. = &c. 
484. Given (237) 

u n t + e sin. u 

required to develope u or any Junction of it according to ike powers ofe. 
Comparing the above form with 

X = ? (t + a X) 
x, t, a, X become respectively 

u, n t, e, sin. u. 
Hence the formula (p) 483. gives 

e 2 d H/(nt)sin. 8 nt* 
+ (u) = -4,(nt) + e -V (n t) sin. n t + - . - ~^ 

e 3 d 2 -4/ (nt) sin. 3 nt} . 

+ 2T3-- n 2 dt 2 - + &C ........ W 

V (n t) being = . 



To farther develope this formula we have generally (see Woodhouse s 
Trig.) 



- 



- 

sin .i (nt) = - ^^ - ; cos. (nt) = 

c being the hyperbolic base, and i any number whatever. Developing the 
second members of these equations, and then substituting 

cos. r n t + V I sin. r n t, and cos. r n t V 1 sin. r n t 
for c rnt ^""S and c~ rn t \ / ~ 1 ., r being any number whatever, we shall 
have the powers i of sin. n t, and of cos. n t expressed in shies and cosines 
of n t and its multiples ; hence we find 

e e 2 

P = sin. n t + -jj sin 2 n t + 5-5 sin. 3 n t + &c. 

^ * O 

= sin. n t 5^5 . {cos. 2 n t 1 } 



in 5 n - 5 sin - 3 n t+ TT2 sill> n 



BOOK I.] NEWTON S PRINCIPIA. 43 

6.5 1 6.5.4 



O Q A t; 05 1 w?w* "^o.^m-p ^.v . 

/i.O.<i.O.U.<w (_ l.-c <& l.iC.o 

&c. 

Now multiply this function by -\J/ (n t), and differentiate each of its 
terms relatively to t a number of times indicated by the power of e which 
multiplies it, d t being supposed constant; and divide these differentials 
by the corresponding power of n d t. Then if P 7 be the sum of the 
quotients, the formula (q) will become 

4 (u) = -^ (n t) + e P . 

By this method it is easy to obtain the values of the angle u, and of 
the sine and cosine of its multiples. Supposing for example, that 

^ u = sin. i u 
we have 

4/ (n t) = i cos. int. 

Multiply therefore the preceding value of P, by i. cos. i n t, and deve- 
lope the product into sines and cosines of n t and its multiples. The 
terms multiplied by the even powers of e, are sines, and those multiplied 
by the odd powers of e, are cosines. We change therefore any term of 
the form, K e T sin. s n t, into + K e 2 r s 2 r sin. s n t, + or obtaining 
according as r is even or odd. In like mariner, we change any term 
of the form, K e 2r + l cos. s n t, into + K e 2r + l . s 2r + l . sin. s n t, or 
-f- obtaining according as r is even or odd. The sum of all these terms 
will be P and we shall have 

sin. i u = sin. i n t + e P . 
But if we suppose 

4/ (u) = u; 
then 

>}/ (n t) = 1 
and we find by the same process 

e 2 
u = n t + e sin. n t -f- ~ ^ . 2 sin. 2 n t 

e 3 
+ .{3 2 sin. 3 n t 3 sin. n t} 

a 4 

. [4, 3 sin. 4 n t 4.2 s sin. 2 n t} 



e s f 54 

34 52 4- |5 4 sin.5nt 5. 3 4 sin. 3 n t+^sin. n 1 



&c. 



44 A COMMENTARY ON [SECT. XL 

a formula which expresses the Excentric Anomaly in terms of the Mean 
Anomaly. 

This series is very convergent for the Planets, Having thus determin 
ed u for any instant, we could thence obtain by means of (237), the cor 
responding values of f and v. But these may be found directly as fol 
lows, also in convergent series. 

485. Required to express g in terms of the Mean Anomaly. 
By (237) we have 

= a (1 e cos. u). 
Therefore if in formula (q) we put 

^ (u) = 1 e cos. u 
we have 

y (n t) = e sin- n t, 
and consequently 

e 3 d sin. 3 n t 
1 e cos. u = 1 e cos. n t + e 2 sin. 2 n t + . j- H &c. 

- 11(11 

Hence, by the above process, we shall find 

P e * e * 

-=-=!+ e cos. n t cos. 2 n t 

a <* 

e 3 
. [3 cos. 3 n t 3 cos. n t } 

<& fit 

.4 2 cos. 4 n t 4. 2 2 . cos. 2 n t} 



2. 3. 2 3 



_ e * . 1 5 3 cos. 5 n t 5. 3 3 cos. 3 n t + ^|. cos. u t j 
_ e& 5 { 6 4 cos. 6 nt 6. 4 4 cos. 4 n t+^|. 2* cos.2nt [ 

*wO i 1 O/i>. 1 iw J 



&c. 

486. To express the True Anomaly in terms of the Mean. 
First we have (237) 

Sin lT ,1+e Sin i 

^r V i e* u 

cos. -g- cos. -g 

.. substituting the imaginary expressions 



C W 

and making 



-i l /1 + e c"^- 1 1. 
-1 + 1 V i e c u v-i+ 1* 

__ e _ 
X ~ 1 + V (1 e ) 



BOOK I.] NEWTON S PRINCIPIA. 45 

we shall have 

1 __ v r . u V 1 

c vV i c u V 
and therefore 



whence expanding the logarithms into series (see p. 28), and putting- 
sines and cosines for their imaginary values, we have 

2 X 2 2 X 3 

v = u + 2 X sin. u -| ^ sin. 2 u -j -- ^ sin. 3 u + &c. 

f o 

But by the foregoing process we have u, sin. u, sin. 2 u, &c. in series 
ordered by the powers of e, and developed into sines and cosines of n t 
and its multiples. There is nothing else then to be done, in order to 
express v in a similar series, but to expand X into a like series. 

The equation, (putting u = 1 + V 1 e 2 ) 



u 



will give by the formula (p) of No. (483) 

1 l i-e 2 ,i(i + 8) e* i (i + 8) (i + 5) e 6 
~i -glT* "2T+T+ 3 2 l + 4 " i 2.3 " <i 6a 

and since 



u = 1 + V I 
we have 



These operations being performed we shall find 
e---j-e 3 + j|j e s | sin. n t 



(103 451 

+ I-96- 6 - 

1097 
+ -960 6 S 

1223 . 



the approximation being carried on to quantities of the order e 6 in 
clusively. 



46 A COMMENTARY ON [SECT. XL 

487. The angles v and n t are here reckoned from the Perihelion ; but 
if we wish to compute from the Aphelion, we have only to make e nega 
tive. It would, therefore, be sufficient to augment the angle n t by r, in 
order to render negative the sines and cosines of the odd multiples of n t ; 
then to make the results of these two methods identical ; we have only in 
the expressions for g and v, to multiply the sines and cosines of odd 
multiples of n t by odd powers of e ; and the even multiples by the even 
powers. This is confirmed, in fact, by the process, a posteriori. 

488. Suppose that instead of reckoning v from the perihelion, we fix 
its origin at any point whatever ; then it is evident that this angle will be 
augmented by a constant, which we shall call =>-, and which- will express 
the Longitude of the Perihelion. If instead of fixing the origin of t at 
the instant of the passage over the perihelion, we make it begin at any 
point, the angle n t will be augmented by a constant which we will call 

e ; and then the foregoing expressions for and v, will become 



a 



= 1 + 4-e 2 (e | e 3 )cos.(nt-H ) ( \ &\ e 4 )cos.2(nt-H 
B 8 o o 



where v is the true longitude of the planet and n t + l its mean longi 
tude, these being measured on the plane of the orbit. 

Let, however, the motion of the planet be referred to a fixed plane a 
little inclined to that of the orbit, and <p be the mutual inclination of the 
two planes, and 8 the longitude of the Ascending Node of the orbit, mea 
sured upon the fixed plane ; also let $ be this longitude measured upon 
the plane of the orbit, so that 6 is the projection of ft and lastly let v, be 
the projection of v upon the fixed plane. Then we shall have 

v, 6, v ft 

making the two sides of a right angled spherical triangle, v /3 being 
opposite the right angle, and <p the angle included between them, and 
therefore by Napier s Rules 

tan. (v, 6) = cos. <p tan. (v /3) ...... (1) 

This equation gives v, in terms of v and reciprocally ; but we can ex 
press either of them in terms of the other by a series very convergent 
after this manner. 

By what has preceded, we have the series 

11 X 2 X 3 

- v = u + X sin. u + ~ sin. 2 u + sin. 3 u + &c. 



BOOK I.] NEWTON S PRINCIPIA 47 

from 



tan 2 v 



by making 



If we change - v into V/ 6 9 and I u into v ft and i~ t-? into 

4 5 J e 

cos. p, we have 



_ cos. p 1 a 

-- - ~~ rfin " _ 

cos. p + 1 If 



The equation between - v and i u will change into the equation be 
tween v, 6 and v ft and the above series will give 
v, 6 = V /3- tan 2 - <p. sin. 2 (v 8) + tan. 4 p. sin. 4 (v /3) 



3 tan. 6 - p sin. 6 (v /3) + &c 



If in the equation between | and ^ , we change ~ v into v _ /3 and 

* /& 

u into v y tf, and , + 6 into ~ , we shall have 

^ 



e cos. <p 



X = tan. 2 - 
and 



v /3 = v/ _^ + tan. z ^ p. sin. 2 (v, 0) 
+ -jg tan. 4 - p. sin. 4 (v y tf) 

+ g tan. 6 - f . sin. 6 ( v/ tf) ..... (4) 

Thus we see that the two preceding series reciprocally interchange, 
ly changing the sign of tan. 2 p, and by changing v, 6, v j3 the (Tne 
for the other. We shall have v/ - t in terms of the sine and cosine of 
n t and its multiples, by observing that we have, by what precedes 

v = n t + + e Q, 

Q being a function of the sine of the angle n t + , - ., and its multi 
ples; and that the formula (i) of number (482) gives, whatever is i, 
sin. i (v /3) = sin. i (n t + s + e Q) 



48 A COMMENTARY ON [SECT. XI. 



Lastly, s being the tangent of the latitude of the planet above the fixed 

plane, we have 

s = tan. <p sin. (v, 6} ; 

and if we call f y the radius-vector projected upon the fixed plane, we 
shall have 



we shall therefore be able to determine v,, s and ^ in converging series 
of the sines and cosines of the angle n t and of its multiples. 

489. Let us now consider very excentric orbits or such as are those of 
the Comets. 

For this purpose resume the equations of No. (237), scil. 



= 



e cos. v 
n t = u e sin. u 



tan. v = 



In this case e differs very little from unity; we shall therefore suppose 

1 e = 
a being very small compared with unity. 

Calling D the perihelion distance of the Comet, we shall have 

D=u(l e) = a a; 
and the expression for g will become 



_ - _ - ____ > 

" 2cos .*JI v a cos.v cos. 2 i- 
which gives, by reduction into a series 



s = 



cos. 2 2 



To get the relation of v to the time t, we shall observe that the expres 
sion of the arc in terms of the tangent gives 
u = 2 tan. i u {l - tan. 2 \ u + \ tan.* I u - &c.} 
But 

1 



BOOK I.] NEWTON S PRINCIPIA. 49 

\ve therefore have 

If 1 / a N 1 1 / \ 2 * 1 1 

u = 2 / -- tan.-v-J i - (- Han.- v-f ~ (^ ) tan. ^v &c.f 

-V 2 a 21 3 \2 a/ 2 5 \2 a/ 2 ) 



Next we have 

2 tan. u 
sin. u = 



1 + tan. 2 u 

A 



i r 

= 2 tan. 4 u 1 1 tan. 2 -^ + tan. 4 \ &c. 
1 25 f. A & 



Whence we get 

/ 

- 



I, 

e sin. u = 2 (1 - *) j tan. - v. 1 - 



Substituting these values of u, and e sin. u in the equation 11 t = u 
e sin. u, we shall have the time t in a function of the anomaly v, by a series 
very convergent ; but before we make this substitution, we shall observe 
that (237) 

n = a ~~ 2 . V m, 
and since 

D = a a, 

we have 

^ 3 
1 D 2 

n 
Hence we find 

5 



* V m 



"> 

If the orbit is parabolic 

a = 
and consequently 

D 



1 

COS. -V 



V m 



{tan. I + Itan. l v} 



which expression may also be got at once from (237). 

The time t, the distance D and sum m of the masses of the sun and 
comet, are heterogeneous quantities, to compare which, we must divide 
each by the units of their species. We shall suppose therefore that the 
mean distance of the sun from the Earth is the unit of distance, so that D 

is expressed in parts of that distance. We may next observe that if T 
VOL. II, D 



50 A COMMENTARY ON [SECT. XI. 

represent the time of a sidereal revolution of the Earth, setting off from 
the perihelion ; we shall have in the equation 

n t = u e sin. u 
u = at the beginning of the revolution, and u = 2 <r at the end of it. 

Hence 

n T = 2 v. 
But we have 

_ 5 

n r= a ? V m = V m, 
\/ m . 

~ rp 

The value of m is not exactly the same for the Earth as for the Comet, 
for in the first case it expresses the sum of the masses of the sun and 
earth ; whereas in the second it implies the sum of the masses of the sun 
and comet : but the masses of the Earth and Comet being much smaller 
than that of the sun, we may neglect them, and suppose that m is the 
same for all Planets and all Comets and that it expresses the mass of the 

2 cr 

sun merely. Substituting therefore for V m its value 7^- in the preced 
ing expression for t ; we shall have 

D*. T f 1 1 3 1 

t = vvnd tan -2 v + s tan - 2 

This equation contains none but quantities comparable with each other ; 
it will give t very readily when v is known ; but to obtain v by means of 
t, we must resolve a Cubic Equation, which contains only one real root. 
We may dispense with this resolution, by making a table of the values of 
v corresponding to those of t, in a parabola of which the perihelion dis 
tance is unity, or equal to the mean distance of the earth from the sun. 
This table will give the time corresponding to the anomaly v, in any par 
abola of which D is the perihelion distance, by multiplying by D ? , the 
time which corresponds to the same anomaly in the Table. We also gel 

the anomaly v corresponding to the time t, by dividing t by D 2 , and 
seeking in the table, the anomaly which corresponds to the quotient 
arising from this division. 

490. Let us now investigate the anomaly, corresponding to the time t, 
in an ellipse of great excentricity. 

If we neglect quantities of the order a \ and put 1 e for a, the above 
expression of t in terms of v in an ellipse, will give 

D * V 2 f tan. v + $ tan. 3 v 

V m ( + (1 e) tan. 2 v f tan. * v - J-tan. + 1 v 
Then, find by the table of the motions of the comets, the anomaly cor- 



BOOK L] NEWTON S PRINCIPIA. 51 

responding to the time t, in a parabola of which D is the perihelion dis 
tance. Let U be this anomaly and U + x the true anomaly in an ellipse 
corresponding to the same time, x being a very small angle. Then if we 
substitute in the above equation U + x for v, and then transform the 
second member into a series of powers of x, we shall have, neglecting the 
square of x, and the product of x by 1 e, 



But by supposition 



tan. U 1 tan. 2 U | tan. 4 1 U} 




U 



Therefore, substituting for x its sine and substituting for sin. 4 i U its 
value (1 cos. 2 1- U) 2 , &c. 

sin. x = T ijy (1 e) tan. | U {4 3 cos. 2 % U 6 cos. 4 \ U} . 

Hence, in forming a table of logarithms of the quantity 
& tan. i U [4, 3 cos. 2 U 6 cos. * \ U} 

it will be sufficient to add the logarithm of 1 e, in order to have that of 
sin. x ; consequently we have the correction of the anomaly U, estimated 
from the parabola, to obtain the corresponding anomaly in a very excen- 
tric ellipse. 

491. To find the masses of such planets as have satellites. 

The equation 

T = 2 ^ a!i 
V m 

gives a very simple method of comparing the mass of a planet, having sa 
tellites, with that of the sun. In fact, M representing the mass of the sun, 
if (t the mass of the planet be neglected, we have 



a 



T _ 

V M 

If we next consider a satellite of any planet ,/, and call its mass p. and 
mean distance from the center of (jf, h, and Tits periodic time, we shall 
have 

T = 2vrh ^ 



2 



_ - 

M a 3 T* 

This equation gives the ratio of the sum of the masses of the planet & 
and its satellite to that of the sun. Neglecting therefore the mass of the 

D2 



52 A COMMENTARY ON [SECT. XI. 

satellite, as small compared with that of the planet, or supposing their ra 
tio known, we have the ratio of the mass of the planet to that of the sun. 

492. To determine the Elements of Elliptical Motion. 

After having exposed the General Theory of Elliptical Motion and 
Method of Calculating by converging series, in the two cases of nature, 
that of orbits almost circular, and the case of orbits greatly excentric, it 
remains to determine the Elements of those orbits. In fact if we call V 
the velocity of /* in its relative motion about M, we have 

V* - dx 2 + dy 2 + dz* 
"dTt^" 

and the last of the equations (P) of No. 478, gives 



To make m disappear from this expression, we shall designate by U 
the velocity which P would have, if it described about M, a circle whose 
radius is equal to the unity of distance. In this hypothesis, we have 

e = a = i, 

and consequently 

U 2 = m. 
Hence 

V 2 = U 

This equation will give the semi-axis major a of the orbit, by means of 
the primitive velocity of p and of its primitive distance from M. But a is 
positive in the ellipse, and infinite in the parabola, and negative in the 
hyperbola. Thus the orbit described by p is an ellipse, a parabola, or hy- 

I 2 
perbola, according as V is < = or > than U ^/ - . It is remarkable 

that the direction of primitive motion has no influence upon the species of 
conic section. 

To find the excentricity of the orbit, we shall observe that if repre 
sents the angle made by the direction of the relative motion of/* with the 
radius-vector, we have 

dp* TT 9 

-T-2-; = V 2 COS. 2 f. 

d t 2 

f 2 I \ 

Substituting for V 2 its value m | J , we have 

d P 2 / 2 1 \ , 

^- t m ( 1 cos. * ; 

d t 1 Vf a / 



BOOK I.] NEWTON S PRINCIPIA. 53 

But by 480 




whence we know the excentricity a e of the orbit. 
To find v or the true anomaly, we have 

a(l e 2 ) 
1 -f e cos - v 

a (1 e 2 ) f 



cos. v 



e f 

This gives the position of the Perihelion. Equations (f ) of No. 480 will 
then give u and by its means the instant of the Planet s passing its peri 
helion. 

To get the position of the orbit, referred to a fixed plane passing 
through the center of M, supposed immoveable, let <p be the inclination of 
the orbit to this plane, and /3 the angle which the radius f makes with the 
Line of the Nodes. Let, Moreover, z be the primitive elevation of /A 
above the fixed plane, supposed known. Then we 
shall have, CAD being the fixed plane, A D the 
line of the nodes, A B = , &c. &c. 

z = B D . sin. p r= sin. (3 sin. p; 
so that the inclination of the oi bit will be known 
when we shall have determined ft. For this pur 
pose, let X be the known angle which the primitive 

direction of the relative motion of /* makes with the fixed plane ; then if 
we consider the triangle formed by this direction produced to meet the 
line of the nodes, by this last line and by the radius f, calling 1 the side 
of the triangle opposite to 8, we have 
, _ g sin. 3 
" sin. (8 + i) 

Next we have 

y = sin. X. 

consequently 

z sin. f 
tan. 8 = 




sin. X z cos. s 

The elements of the Planetary Orbit being determined by these formu 
las, in terms of and z, of the velocity of the planet, and of the direction 
of its motion, we can find the variation of these elements corresponding 

D3 



54 A COMMENTARY ON [SECT. XL 

to the supposed variations in the velocity and its direction; and it will be 
easy, by methods about to be explained, from hence to obtain the differ 
ential variations of the Elements, due to the action of perturbing forces. 
Taking the equation 

V 2 = U 2 {- - 1 }. 
I g a J 

In the circle a = g and .*. 

V = U J- 

\ g 

so that the velocities of the planets in different circles are reciprocally as 
the squares of their radii (see Prop. IV of Princip.) 
In the parabola, a = oo , 



_ 

the velocities in the different points of the orbit, are therefore in this case 
reciprocally as the squares of the radius- vectors ; and the velocity at each 
point, is to that which the body would have if it described a circle whose 
radius = the radius-vector g, as V 2 : 1 (see 160) 

An ellipse indefinitely diminished in breadth becomes a straight line, 
and in this case V expresses the velocity of /*, supposing it to descend in 
a straight line towards M. Let A* fall from rest, and its primitive dis 
tance be g ; also let its velocity at the distance g be V ; the above expres 
sion will give 



whence 

V = U J -^^- 

V g/ 

Many other results, which have already been determined after another 
manner, may likewise be obtained from the above formula. 
493. The equation 

_dx 



_ 

dt 2 

is remarkable from its giving the velocity independently of the excentricity. 
It is also shown from a more general equation which subsists between the 
axis-major of the orbit, the chord of the elliptic arc, the sum of the ex 
treme radius-vectors, and the time of describing this arc. 
To obtain this equation, we have 
a(l e 2 ) 
1 + e cos. v 



BOOK I.] NEWTON S PRINCIP1A. 55 

g = a (1 e cos. u) 

3L 

t == a J (u e sin. a) ; 

in which suppose f, v, u, and t to correspond to the first extremity of the 
elliptic arc, and that p , v , u , t belong to the other extremity ; so that we 
also have 



1 + e cos - v/ 
P = a ( 1 e cos. u ) 

t = a 2 (u e sin. u ). 
Let now 

_t - T- u/ ~ u - 8- 

t " L J. 9 A ^ f J 9 

H-_^ = ; g + s = R; 

then, if we take the expression oft from that oft , and observe that 

sin. u sin. u = 2 sin. 8 cos. 8 
we shall have 

T = 2 a * ?jS e sin. jS cos. 8}. 
If we add them together taking notice that 

cos. u + cos. u = 2 cos. 8. cos. 8 
we shall get 

R = 2 a (1 e cos. 8 cos. /3 ). 

Again, if c be the chord of the elliptic arc, we have 

C 2 =rf 2 + f /2 2pf COS. (v v ) 

but the two equations 

P =. -. \ ; P a (1 e cos. u) 

1 4~ e cos - v 

give these 



cos. u e . aVl e 2 . sin. u 
cos. v = a ; sn>. v = 

s e 

and in like manner we have 



cos. u e , a V 1 e" sin. u 
cos. v = a . - f ; sin. v = , ; 

whence, we get 

g / cos. (v v ) = a 2 (e cos. u) (e cos. u ) +a 2 (1 e ! ) sin. u sin. u ; 

and consequently 

c ! = 2a 2 (l e 2 ) 1 sin. u sin. u cos. u cos. u \ 
4- a 8 e 2 (cos. u cos. u ) * ; 
D 4 



56 A COMMENTARY ON [SECT. XI. 

But 

sin. u sin. u + cos. u cos. u = 2 cos. * /3 1 
cos. u cos. u = 2 sin. /3 sin. /3 
.-.c 2 = 4 a s sin. 8 /3(l e 2 cos. 2 /30- 
We therefore have these three equations, scil. 
R = 2 a { 1 e cos. |8 cos. } ; 
jT = 2 a ^ Jj3 _ e sin. j3 cos. /3 } , 
c 2 = 4a 2 sin. 2 (1 e*cos. 2 /3). 
The first of them gives 

a , 2 a R 

ef*/~\O /*v ~~ , __ 
V-WO* ^ _ >- 

2 a cos. p 
and substituting this value of e cos. ft in the two others, we shall have 



2 



c 2 = 4a 2 tan. z /3|cos. 2 /3 ( 2 



These two equations do not involve the excentricity e, and if in the 
first we substitute for (S its value given by the second, we shall get Tina 
function c, R, and a. Thus we see that the time T depends only on the 
semi-axis major, the chord c and the sum R of the extreme radius- 
vectors. 
If we make 

2 a R + c , _ 2 a R c 

~2lT ~^~a~~ 

the last of the preceding equations will give 



cos. 2/3 = 22 + V (1 z 2 ) . (1 2 2 ); 
whence 

2 j3 = cos. - z f cos. - 2 

(for cos. (A B) = cos. A cos. B + sin. A sin. B). 
Consequently 

sin. (cos.- 1 z ) sin. (cos. - I z) 
tan. /3 = - z + z , 

we have also 

2 a R 

2 + 2 =^. 

Hence the expression of T will become, observing that if T is the du 
ration of the sidereal revolution, whose mean distance from the sun is 
taken for unity, we have 



BOOK I.] NEWTON S PRINCIPJA. 57 

T - 2cr, 

a l T 
T = -g Jcos.- 1 z cos.- z sin. (cos.- 4 z ) + sin.(cos~ 1 z)j ... (a) 

Since the same cosines may belong to many arcs, this expression is 
ambiguous, and we must take care to distinguish the arcs which corre 
spond to z, z . 

In the parabola, the semi-axis major is infinite, and we have 

If , A 1 /R + C\ f 

cos. ~ l z sin. (cos. z ) = 3? I - ) . 

6 \ a / 

And making c negative we shall have the value of 

cos. ~" 1 z sin. (cos. 1 z) ; 

hence the formula (a) will give the time T employed to describe the arc 
subtending the chord c, scil. 

T = wzfo *< + c >*+fe + * - ) f ? ; 

the sign being taken, when the two extremities of the parabolic arc are 
situated on the same side of the axis of the parabola. 
Now T being = 365.25638 days, we have 

~ = 9. 688754 days. 

12 v J 

The formula (a) gives the time of a body s descent in a straight line to 
wards the focus, beginning from a given distance; for this, it is suffi 
cient to suppose the axis-minor of the ellipse indefinitely diminished. If 
we suppose, for example, that the body falls from rest at the distance 2 a 
from the focus and that it is required to find the time (7") of falling to 
the distance c, we shall have 

R = 2 a + , f = 2 a c 
whence 



z = _ 1, z = - 

a 

and the formula gives 

a * T ( , c a / 2 a c c\ 

T = \9 cos. ~ -- f- . / - 5 f . 

2 T I a \ a 2 

There is, however, an essential difference between elliptical motion to 
wards the focus, and the motion in an ellipse whose breadth is indefinite 
ly small. In the first case, the body having arrived at the focus, passes 
beyond it, and again returns to the same distance at which it departed ; 
but in the second case, the body having arrived at the focus immediately 
returns to the point of departure. A tangential velocity at the aphelion, 



58 A COMMENTARY ON [SECT. XL 

however small, suffices to produce this difference which has no influence 
upon the time of the body s descent to the center, nor upon the ve 
locity resolved parallel to the axis-major. Hence the principles of the 
7th Section of Newton give accurately the Times and Velocities, although 
they do not explain all the circumstances of motion. For it is clear that 
if there be absolutely no tangential velocity, the body having reached the 
center offeree, will proceed beyond it to the same distance from which it 
commenced its motion, and then return to the center, pass through it, 
and proceed to its first point of departure, the whole being performed in 
just double the time as would be required to return by moving in the in 
definitely small ellipse. 

494. Observations not conducting us to the circumstances of the pri 
mitive motion of the heavenly bodies ; by the formulas of No. 492 we 
cannot determine the elements of their orbits. It is necessary for this 
end to compare together their respective positions observed at different 
epochs, which is the more difficult from not observing them from the 
center of their motions. Relatively to the planets, we can obtain, by 
means of their oppositions and conjunctions, their Heliocentric Longitude. 
This consideration, together with that of the smallness of the excentricity 
and inclination of their orbits to the ecliptic, affords a very simple method 
of determining their elements. But in the present state of astronomy, 
the elements of these orbits need but very slight corrections ; and as the 
variations of the distances of the planets from the earth are never so great 
as to elude observation, we can rectify, by a great number of observations, 
the elements of their orbits, and even the errors of which the observa 
tions themselves are susceptible. But with regard to the Comets, this is 
not feasible ; we see them only near their perihelion : if the observations 
we make on their appearance prove insufficient for the determination of 
their elements, we have then no means of pursuing them, even by thought, 
through the immensity of space, and when after the lapse of ages, they 
again approach the sun, it is impossible for us to recognise them. It be 
comes therefore important to find a method of determining, by observa 
tions alone during the appearance of one Comet, the elements of its orbit. 
But this problem considered rigorously surpasses the powers of analysis, 
and we are obliged to have recourse to approximations, in order to obtain 
the first values of the elements, these being afterwards to be corrected to 
any degree of accuracy which the observations permit. 

If we use observations made at remote intervals, the eliminations will 
lead to impracticable calculations ; we must therefore be content to con- 



BOOK I.] NEWTON S PRINCIPIA. 59 

sider only near observations ; and with this restriction, the problem is abun 
dantly difficult. 

It appears, that instead of directly making use of observations, it is 
better to get from them the data which conduct to exact and simple re 
sults. Those in the present instance, which best fulfil that condition, are 
the geocentric longilude and latitude of the Comet at a given instant, and 
their first and second differences divided by the corresponding powers of 
the element of time ; for by means of these data, we can determine rigo 
rously and with ease, the elements, without having recourse to a single 
integration, and by the sole consideration of the differential equations of 
the orbit. This way of viewing the problem, permits us moreover, to 
employ a great number of near observations, and to comprise also a con 
siderable interval between the extreme observations, which will be found 
of great use in diminishing the influence of such errors, as are due to ob 
servations from the nebulosity by which Comets are enveloped. Let us 
first present the formulas necessary to obtain the first differences, of the 
longitude and latitude of any number of near observations ; and then de 
termine the elements of the orbit of a Comet by means of these differences ; 
and lastly expose the method which appears the simplest, of correcting 
these elements by three observations made at remote intervals. 

495. At a given epoch, let a be the geocentric longitude of a Comet, 
and d its north geocentric latitude, the south latitudes being supposed ne 
gative. If we denote by s, the number of days elapsed from this epoch, 
the longitude and latitude of the Comet, after that interval, will, by using 
Taylor s Theorem (481), be expressed by these two series 

d ax s * /d " \ 



We must determine the values of 

/d a s /d 2 a\ /d rf\ 

" ld*>? (dT*)> &c " (ds) &c> 

by means of several observed geocentric longitudes and latitudes. To do 
this most simply, consider the infinite series which expresses the geocen 
tric longitude. The coefficients of the powers of s, in this series, ought to 
be determined by the condition, that by it is represented each observed 
longitude; we shall thus have as many equations as observations; and i( 
their number is n, we shall be able to find from them, in series, the n 



60 A COMMENTARY ON [SECT. XL 

quantities , f-r ) , &c. But it ought to be observed that s being sup 
posed very small, we may neglect all terms multiplied by s n , s n + l , &c. 
which will reduce the infinite series to its n first terms ; which by n ob 
servations we shall be able to determine. These are only approximations, 
and their accuracy will depend upon the smallness of the terms which are 
omitted. They will be more exact in proportion as s is more diminutive, 
and as we employ a greater number of observations. The theory of inter 
polations is used therefore To find a rational and integer function qfs such, 
that in substituting therein for s the number of days which correspond to each 
observation, it shall become the observed longitude. 

Let (3, /3 , f3" f &c. be the observed longitudes of the comet, and by 
i, i , i", &c. the corresponding numbers of days from the given epoch, the 
numbers of the days prior to the given epoch being supposed negative. 
If we make 

R R R" R> R " R 

p a B " - 3 fi " 
" P ) // / OP 

>l!f - tf> -> 



1" 1 1 

y d 2 ^ 



; &c. 



i " i 

&c.; 
the required functions will be 

for it is easy to perceive that if we make successively s = i, s = i , s = i", &c. 
it will change itself into /3, /3 , /3 /x , &c. 

Again, if we compare the preceding function with this 



we shall have by equating coefficients of homogeneous terms. 

i . 6 2 /3 i . i . \" 



&c. 



The higher differences of a will be useless. The coefficients of these 
expressions are alternately positive and negative ; the coefficient of d r 13 
is, disregarding the sign, the product of r and r together of r quantities 
i, i , . . . . i (r - 1! in the value of ; it is the sum of the products of the 



BOOK L] NEWTON S PRINCIPIA. 

same quantities, r 1 together in the value of (-r 

of the products of these quantities r 2, together in the value of 



^d s 2 > 
If 7, 7 , 7", &c. be the observed geocentric latitudes, we shall have the 

values of d, (-r ) , (-1 2 ) > &c. by changing in the preceding expressions 

for a (-p) 5 ( i ") 5 &c. the quantities /3, (3 , /3" into 7, /, 7". 

These expressions are the more exact, the greater the number of ob 
servations and the smaller the intervals between them. We might, 
therefore, employ all the near observations made at a given epoch, pro 
vided they were accurate; but the errors of which they are always sus 
ceptible will conduct to imperfect results. So that, in order to lessen the 
influence of these errors, we must augment the interval between the ex 
treme observations, employing in the investigation a greater number of 
them. In this way with five observations we may include an interval of 
thirty-five or forty degrees, which would give us very near approximations 
to the geocentric longitude and latitude, and to their first and second 
differences. 

If the epoch selected were such, that there were an equal number of 
observations before and after it, so that each successive longitude may 
have a corresponding one which succeeds the epoch. This condition will 

give values still more correct of a, f-t J and ( ,- -) j an( ^ it easily appears 

that new observations taken at equal distances from either side of the epoch, 
would only add to these values, quantities which, with regard to their last 

i g 

terms, would be as s 2 ( -. 2 j to . This symmetrical arrangement takes 

place, when all the observations being equidistant, we fix the epoch at 
the middle of the interval which they comprise. It is therefore advanta 
geous to employ observations of this kind. 

In general, it will be advantageous to fix the epoch near the middle of 
this interval ; because the number of days included between the extreme 
observations being less considerable, the approximations will be more con 
vergent. We can simplify the calculus still more by fixing the epoch at 
the instant of one of the observations ; which gives immediately the values 
of , and 6. 



62 A COMMENTARY ON [SECT. XI 

When we shall have determined as above the values of 
d\ /d 2 \ /d S\ i /d 2 



\ / \ i / \ 

T 2 ) GB)I and (dp) 



we shall then obtain as follows the first and second differences of a, and fl 
divided by the corresponding powers of the elements of time. If we neg 
lect the masses of the planets and comets, that of the sun being the unit 
of mass ; if, moreover, we take the distance of the sun from the earth for 
the unit of distance ; the mean motion of the earth round the sun will 
be the measure of the time t. Let therefore X be the number of se 
conds which the earth describes in a day, by reason of its mean sidereal 
motion ; the time t corresponding to the number of days will be X s ; we 
shall, therefore, have 

(d \ 1 /d \ 
d~~t/ " T \dl) 

(d 2 a\ 1 /d 2 a\ 
d"tV ~ X~Hd sV* 

Observations give by the Logarithmic Tables, 

log. X = 4. 0394622 
and also 

log. X 2 = log. X + log. -g 
R bein the radius of the circle reduced to seconds ; whence 






log. X s = 2.2750444; 

J J 2 

.-. if we reduce to seconds, the values of (-p) 5 and of (-T 2 ) , we shall 



1 



have the logarithms of ( ,-") , and of (^-^) by taking from the logarithms 

*C1 t Cl I / 

of these values the logarithms of 4. 039422, and 2. 2750444. In like 
manner we get the logarithms of ( r-V ( T .4) , after subtracting the 



same logarithms, from the logarithms of their values reduced to seconds. 
On the accuracy of the values of 

d 



depends that of the following results ; and since their formation is very 
simple, we must select and multiply observations so as to obtain them with 
the greatest exactness possible. We shall determine presently, by means 
of these values, the elements of the orbit of a Comet, and to generalize 
these results, we shall 



BOOK I.] NEWTON S PRINCIPIA. 63 

496. Investigate the motion of a system of bodies sollicited by any forces 
whatever. 

Let x, y, z be the rectangular coordinates of the first body ; x , y , z 
tliose of the second body, and so on. Also let the first body be sollicited 
parallel to the axes of x, y, z by the forces X, Y, Z, which we shall sup 
pose tend to diminish these variables. In like manner suppose the second 
body sollicited parallel to the same axes by the forces X , Y , Z , and so 
on. The motions of all the bodies will be given by differential equations 
of the second order 



&c. = &c. 

If the number of the bodies is n, that of the equations will be 3 n ; and 
their finite integrals will contain 6 n arbitrary constants, which will be the 
elements of the orbits of the different bodies. 

To determine these elements by observations, we shall transform the 
coordinates of each body into others whose origin is at the place of the 
observer. Supposing, therefore, a plane to pass through the eye of the 
observer, and of which the situation is always parallel to itself, whilst the 
observer moves along a given curve, call r, r r", &c. the distances of 
the observer from the different bodies, projected upon the plane ; 
, a , a", &c. the apparent longitudes of the bodies, referred to the same 
plane, and 6, ff, 0", &c. their apparent latitudes. The variables x, y, z 
will be given in terms of r, , 0, and of the coordinates of the observer. 
In like manner, x , y , z will be given in functions of r 7 , a , ff, and of the 
coordinates of the observer, and so on. Moreover, if we suppose that the 
forces X, Y, Z ; X 7 , Y , Z , &c. are due to the reciprocal action of the 
bodies of the system, and independent of attractions ; they will be given in 
functions of r, r , r", &c. ; a, a , a", &c. ; 6, 6 , 6", &c. and of known quan 
tities. The preceding differential equations will thus involve these new 
variables and their first and second differences. But observations make 
known, for a given instant, the values of 

/d ax /d*\ . /d 0\ /d 2 6\ , /da \ 

*> (ai) (arO Men) (dT*) ; " Car) &c - 

There will hence of the unknown quantities only remain r, r 7 , r", &c. 
and their first and second differences. These unknowns are in number 
3 n, and since we have 3 n differential equations, we can determine them. 



64 A COMMENTARY ON [SECT. XI. 

At the same time we shall have the advantage of presenting the first and 
second differences of r, r , r", &c. under a linear form. 

The quantities , 6, r, , ^, r 7 , &c. and their first differences divided by 
d t, being known ; we shall have, for any given instant, the values of 
x, y, z, x , y , z , &c. and of their first differences divided by d t. If we 
substitute these values in the 3 n finite integrals of the preceding equa 
tions, and in the first differences of these integrals ; we shall have 6 n 
equations, by means of which we shall be able to determine the 6 n arbi 
trary constants of the integrals, or the elements of the orbits of the dif 
ferent bodies. 

497. To apply this method to the motion of the Comets, 

We first observe that the principal force which actuates them is the 
attraction of the sun ; compared with which all other forces may be ne 
glected. If, however, the Comet should approach one of the greater 
planets so as to experience a sensible perturbation, the preceding method 
will still make known its velocity and distance from the earth ; but this 
case happening but very seldom, in the following researches, we shall ab 
stain from noticing any other than the action of the sun. 

If the sun s mass be the unit, and its mean distance from the earth the 
unit of distance; if, moreover, we fix the origin of the coordinates 
x, y, z of a Comet, whose radius-vector is g ; the equations (0) of No. 475 
will become, neglecting the mass of the Comet, 



o _ , 
2 



(k) 



dt 2 

Let the plane of x, y be the plane of the ecliptic. Also let the axis of 
x be the line drawn from the center of the sun to the first point of aries, 
at a given epoch ; the axis of y the line drawn from the center of the sun 
to the first point of cancer, at the same epoch ; and finally the positive 
values of z be on the same side as the north pole of the ecliptic. Next 
call x , y 7 the coordinates of the earth and R its radius-vector. This be 
ing supposed, transfer the coordinates x, y, z to others relative to the 
observer ; and to do this let a be the geocentric longitude, and r its dis 
tance from the center of the earth projected upon the ecliptic ; then we 
shall have 

x = x -f- r cos. ; y = y + r sin. a; z = r tan. 6. 



BOOK I.] NEWTON S PRINCIPIA. 66 

If we multiply the first of equations (k) by sin. a, and take from the re 
sult tlve second multiplied by cos. a, we shall have 

d 2 x d 2 y x sin. a y cos. a 

whence we derive, by substituting for x, y their values given above, 
d 2 x d y x sin. a y cos. a 



= s.. Trr -. - 

d r\ /da 



The earth being retained in its orbit like a comet, by the attraction of 
the sun, we have 



d l x , *_ n _ d V , jr 
dt 2 + R S} ~dTt 2 + R S; 



which give 

We shall, therefore, have 



d 2 x d 2 V y cos. a x sill, a 

sin. a - cos. a . .-. v = = =^ 

d t z dt* R 3 



n / da /d 2 a 

- 



Let A be the longitude of the earth seen from the sun ; we shall have 

x = R cos. A ; y = R sin. A ; 
therefore 

y cos. a x sin. a = R sin. (A a) ; 
and the preceding equation will give 

/dx 

/drx Rsin.(A a) M 1) \d tV 

Vdt/ = /dUx (R 3 ""^/- -TJ- 

- 



Now let us seek a second expression for (j~\ . For this purpose we 

will multiply the first of equations (k) by tan. & . cos. , the second by 
tan. 6 sin. a, and take the third equation from the sum of these two pro 
ducts ; we shall thence obtain 



sn . 



\- tan 6 x cos - a + y sin * 

3 



_ _ _ 
S 3 ~dt 2 g 3 

This equation will become by substitution for x, y, z 

./ /d 2 x / , x\ , /d 2 y y\ . \ 

= tan. l( (^ + ^) cos. + (^ + ) sin. } 

VOL. TI. K 



66 



A COMMENTARY ON [SECT. XI. 



But 

,.- = co, 



= R cos . ( A ) g- 



Therefore, 




R sin. 6 cos. cos. (A ) / 1 1 ) / 2 \ 

+ - -TdV -tr"R 3 / 

Vd J 

If we take this value of (^) from the first and suppose 



sin. tf cos. tf cos. (A ) + ( ) sin. (A - ) 
we shall have 



The projected distance r of the comet from the earth, being always po 
sitive, this equation shows that the distance s of the comet from the sun, 
is less or greater than the distance R of the sun from the earth, according 
as (i! is positive or negative; the two distances are equal if (if = 0. 

By inspection alone of a celestial globe, we can determine the sign of 
// ; and consequently whether the comet is nearer to or farther from the 
Earth. For that purpose imagine a great circle which passes through 
two Geocentric positions of the Comet infinitely near to one another. 
Let 7 be the inclination of this circle to the ecliptic, and X the longitude 
of its ascending node ; we shall have 

tan. 7 sin. (a X) = tan. 6 ; 

\vricncc 

d 6 sin. (a X) = a a sin. 6 cos. 6 cos. (a X). 



BOOK I.] NEWTON S PRINCIPIA. 67 

Differentiating, we have, also 

/dav/d*0\ /d<K/d 2 a\ ( /dax/d0\ 8 

= (di) (jtO-VD Grrv + 2 v d i) (di) lan - 

I 3 



+ ( v-r) 



sn. cos. 



d 2 0, being the value of d 2 6, which would take place, if the apparent mo 
tion of the Comet continued in the great circle. The value of y! thus be 
comes, by substituting for d 6 its value 

d a sin. 6 cos. 6 cos. (a X) 
sin. (a X) 



sin. cos. sin. (A X) 

The function . V - .- is constantly positive : the value of IL is there- 
sin. 6 cos. 6 

c i- /d 2 6 \ /d ^Ai. 

k>re positive or negative, according asf-j ^J (-p-pjhas the same or 

a different sign from that of sin. (A X). But A X is equal to two 
right angles plus the distance of the sun from the ascending node of the 
great circle. Whence it is easy to conclude that fjf will be positive or 
negative, according as in a third geocentric position of the comet, inde 
finitely near to the two first, the comet departs from the great circle on 
the same or the opposite side on which is the sun. Conceive, therefore, 
that we make a great circle of the sphere pass through the two geocentric 
positions of the comet ; then according as, in a third consecutive geocen 
tric position, the comet departs from this great circle, on the same side as 
the sun or on the opposite one, it will be nearer to or farther from the 
sun than the Earth. If it continues to appear in this great circle, it will 
be equally distant from both ; so that the different deflections of its ap 
parent path points out to us the variations of its distance from the sun. 

To eliminate from equation (3), and to reduce this equation so as to 
contain no other than the unknown r, we observe that g 2 = x z + y 2 -f- z* 
in substituting for x, y, z, their values in terms of 

r, a, and ; 
and we have 

S a - = x 2 + y /2 + 2rx cos. a + y sin. a] + ^ J 

but we have 

x R cos. A, y = R sin. A ; 

c^ + 2 R r cos - < A - a) + Il ; 

E2 



68 A COMMENTARY ON [SECT. XL 

But 

x = R cos. A ; y = R sin. A 

.-. P 2 = -^r, + 2 R r cos. (A ) + tt 2 - 
cos. 2 6 

If we square the two members of equation (3) put under this form 

e*{p R 2 r + 1}= R 3 
we shall get, by substituting for g 2 , 

/ -^- -f 2 R r cos. (A ) + R 2 j .{(* R 2 r + l} = R c . . . (4) 
\ cos. 2 6 ) 

an equation in which the only unknown quantity is r, and which will rise 
to the seventh degree, because a term of the first member being equal to 
R 6 , the whole equation is divisible by r. Having thence determined r, 

we shall have (-. ) by means of equations (1) and (2). Substituting, for 

example, in equation (1), for 3 R -, its value -~ , given by equation 
(3) ; we shall have 



The equation (4) is often susceptible of many real and positive roots ; 
reducing it and dividing by r, its last term will be 

2 R 5 cos. 6 W R 3 + 3 cos. (A a)}. 

Hence the equation in r being of the seventh degree or of an odd de 
gree, it will have at least two real positive roots if [if R 3 + 3 cos. (A a) 
is positive; for it ought always, by the nature of the problem, to have 
one positive root, and it cannot then have an odd number of positive 
roots. Each real and positive value of r gives a different conic section, 
for the orbit of the comet ; we shall, therefore, have as many curves 
which satisfy three near observations, as r has real and positive values ; 
and to determine the true orbit of the comet, we must have recourse to a 
new observation. 

498. The value of r, derived from equation (4) would be rigorously 

exact, if 



were exactly known ; but these quantities are only approximate. In fact, 
by the method above exposed, we can approximate more and more, mere 
ly by making use of a great number of observations, which presents the 
advantage of considering intervals sufficiently great, and of making the 
errors arising from observations compensate one another. But this 



BOOK I.] NEWTON S PUINCIPIA. 69 

method has the analytical inconvenience of employing more than three 
observations, in a problem where three are sufficient. This may be 
obviated, and the solution rendered as approximate as can be wished by 
three observations only, after the following manner. 

Let a and 6, representing the geocentric longitude and latitude of the 
intermediate ; if we substitute in the equations (k) of the preceding 
No. instead of x, y, z their values x + r cos. a ; y + r sin. a ; and 

r tan. 6; they will give (_. 2 V ( i 1 2 ) anc ^ ("rT 2 ) m ^ unc ^ ons f r > "> and 
0, of their first differences and known quantities. If we differentiate these, 
we shall havef-. j} , (-^ 5} and (-T 3 } in terms of r, a, 6, and of their 

first and second differences. Hence by equation (2) of 497 we may eli 
minate the second difference of r by means of its value and its first differ 

ence. Continuing to differentiate successively the values of (-r ) > (--, 3 ) > 

and eliminating the differences of a, and of superior to second differences, 
and all the differences of r, we shall have the values of 
d 



&Ct in terms ot 
d d /d 2 tfv 



this being supposed, let 

/> a, a , 

be the three geocentric observed longitudes of the Comet; /3 0, tf its 
three corresponding geocentric latitudes; let i be the number of days 
which separate the first from the second observation, and i the interval 
between the second and third observation ; lastly let X be the arc which 
the earth describes in a day, by its mean sidereal motion ; then by (481) 
we have 

. . /d \ , i 2 . X 2 /d 2 \ i 3 X 3 fd 3 \ 

"< = - Hen) + TTW- naCfw + &c - ; 

, , ., , /da\ i /2 . X 2 /d 2 a x i /3 . X 3 /d 3 \ 

= + 1 . x ( d - t ) + L 2 (^ + 1^3 ( a T3 ) + &c. , 



/v .x 2 / 2 ^ 
Cdl) + 172- (dl -0 



2 



70 A COMMENTARY ON [Stcx. XI. 

If we substitute in these series for 



their values obtained above, we shall have four equations between the 
five unknown quantities 



These equations will be the more exact in proportion as we consider a 
greater number of terms in the series. We shall thus have 



/\ / a\ / \ / \ 
\d~t)> \2TtV! vTt/ VdT 2 ,) 



in terms of r and known quantities; and substituting in equation (4) of 
the preceding No. it will contain the unknown r only. As to the rest, 
this method, which shows how to approximate to r by employing three 
observations only, would require in practice, laborious calculations, and 
it is a more exact and simple process to consider a greater number of ob 
servations by the method of No, 495. 

499. When the values of r and f-i~J shall be determined, we shall have 

those of 

/d x\ /d y\ , /d z\ 

x > McTt) (dt) and (dl) 

by means of the equations 

x = R cos. A + r cos. a 
y = R sin. A + r sin. 
z = r tan. 6 

and of their differentials divided by d t, viz. 
dx\ /d R\ T, /d A\ . /d r\ 

s - A - Rs n - A + cos - 



/ r\ 

- A + (ai) 



v\ /d R\ . ,, /d A\ 

t ) = ( dnr) sm - A + R ( d v) cos - A 




d t 



The values of ( ( \ A ) and of (A) are given by the Theory of the 

motion of the Eai th : 

To facilitate the investigation let E be the excentricity of the earth s 



BOOK I.] NEWTON S PRINCIPIA. 71 

orbit, and H the longitude of its perihelion; then by the nature of 

elliptical motion we have 

/dAx V(i-E 2 ). _ 1-E 2 

VdT/ ~U*~ 1 + Ecos. (A H)* 

These two equations give 

/d Rx E sin. (A H) 
Idt/ : ~V (1 E 2 ) 

Let R be the radius- vector of the earth corresponding to the longitude 
A of this planet augmented by a right angle ; we shall have 



___ _ 

1 E sin. ( A H) 

whence is derived 

T-, /A Tjv R 1 + 

E sin. ( A H) = - -: 



/d Rx R / + E _-- 1 

\dt) ~ R V (1 E 2 ) 

If we neglect the square of the excentricity of the earth s orbit, which is 
very small, we shall have 

/d A\ _ 1 /d Rx , 

ITF)-R- ; (dr) = R - 

the preceding values of (T~T) and f -p 2 " Will hence become 

d xx _ sin. A , /d r\ /d\ 

)cos - A - ~ir + (di) cos - a - r (di) sln - a; 

- cos - A /d 



fy\ /-of i\ - 
(df) =( R - ^ Sm 

R, R , and A being given immediately by the tables of the sun, the esti 
mate of the six quantities x, y, z, ( j- ^) (d~?) (d?) wil1 be 



when r and - shall be known. Hence we derive the elements of the 



orbit of the comet after this mode. 

The indefinitely small sector, which the projection of the radius- vector 
and the comet upon the plane of the ecliptic describes during the element 

of time d t, is - 21_XJ - and it is evident that this sector is posi 
tive or negative, according as the motion of the comet is direct or retro 
grade. Thus in forming the quantity x (jl) _ y (1~), it will indicate 

by its sign, the direction of the motion of the comet. 

E 4, 



72 A COMMENTARY ON [SECT. XI. 

To determine the position of the orbit, call <p its inclination to the 
ecliptic, and I the longitude of the node, which would be ascending if the 
motion of the comet were direct or progressive. We shall have 
z = y cos. I tan. <p x sin. I tan. <p 

These two equations give 



tan. I = 



tan. = 



Wherein since <p ought always to be positive and less than a right 
angle, the sign of sin. I is known. But the tangent of I and the sign of 
its sine being determined, the angle I is found completely. This angle 
is the longitude of the ascending node of the orbit, if the motion is pro 
gressive; but to this we must add two right angles, in order to get the 
longitude of the node when the motion is retrograde. It would be more 
simple to consider only progressive motions, by making vary p, the in 
clination of the orbits, from zero to two right angles ; for it is evident that 
then the retrograde motions correspond to an inclination greater than a 



right angle. 






In this case, tan. <p has the same sign as x ( j^-) y (-i ) > which will 

determine sin. I, and consequently the angle I, which always expresses 
the longitude of the ascending node. 

If a, a e be the semi-axis major and the excentricity of the orbit, we 
have (by 492) in making m = 1, 



The first of these equations gives the semi-axis major, and the second 
the excentricity. The sign of the function x (j^) + ? (j~D + z (dl) 
shows whether the comet has already passed its perihelion ; for it ap 
proaches if this function is negative; and in the contrary case, the comet 
recedes from that point. 



BOOK I.] NEWTON S PRINCIPIA. 73 

Let T be the interval of time comprised between the epoch and pas 
sage of the comet over the perihelion; the two first of equations (f) (480) 

_5 

will give, observing that m being supposed unity we have n = a 2 , 
= a (1 e cos. u) 

5 

T = a 2 (u e cos. u). 

The first of these equations gives the angle u, and the second T. This 
time added to or subtracted from the epoch, according as the comet ap 
proaches or leaves its perihelion, will give the instant of its passage over 
this point. The values of x, y, determine the angle which the projection 
of the radius-vector makes with the axis of x ; and since we know the an 
gle I, formed by this axis and by the line of the nodes, we shall have the 
angle which this last line forms with the projection of g ; whence we derive by 
means of the inclination p of the orbit, the angle formed by the line of the 
nodes and the radius f. But the angle u being known, we shall have by 
means of the third of the equations (f), the angle v which this radius forms 
with the line of the apsides ; we shall therefore have the angle comprised 
between the two lines of the apsides and of the nodes, and consequently, 
the position of the perihelion. All the elements of the orbit will thus be 
determined. 

500. These elements are given, by the preceding investigations, in terms 

of r, (-1-7) and known quantities ; and since (-,- ) is given in terms of r 

by No. 497, the elements of the orbit will be functions of r and known 
quantities. If one of them were given, we should have a new equation, 
by means of which we might determine r ; this equation would have a 
common divisor with equation (4) of No. 497; and seeking this di 
visor by the ordinary methods, we shall obtain an equation of the first 
degree in terms of r ; we should have, moreover, an equation of condition 
between the data of the observations, and this equation would be that 
which ought to subsist, in order that the given element may belong to the 
orbit of the comet. 

Let us apply this consideration to the case of nature. First suppose 
that the orbits of the comets are ellipses of great excentricity, and are 
nearly parabolas, in the parts of their orbits in which these stars are 
visible. We may therefore without sensible error suppose a = <x>, and 

consequently - = 0; the expression for - of the preceding No. will there 
fore give 



74 A COMMENTARY ON [SECT. XI. 

2 dx 2 + dy 2 + dz 2 

: 7 u dt 2 

If we then substitute for f-r\ IT-M and (-5) their values found in 
vd tJ \d t/ \d t/ 



the same No., we shall have after all the reductions and neglecting the 
square of R 1, 

- (So* ( 



cos. 



2, 



Substituting in this equation for (-5 ) its value 

J/ d2 \ , xl 
i ( -5 5- ) + /* sin. (A a) f , 

ax I \d t 2 / ^ J 



found in No. 497, and then making 

./d\ z T, . /d\ 4 . f /d 2 a\ , . . /A N ) " 

Hal) B = Hai) + 1 (err*) + " sin - < A - "U 

C /d \ /d ^\ ^ 2 

J tan. 6. (-, - a } + ij, tan. 6 sin. (A ) 4 

v. \ct t / cos. 9 J 

and 

C = d t -JL /5El^ZI^__(R __i)cos. (A )l 

/d\ ( K J 



we shall have 



= Br 2 + Cr + ^i ~ 



and consequently 

r*+ Cr + - 2 =: 4. 



This equation rising only to the sixth degree, is in that respect, more 



BOOK I.] NEWTON S PRINCIPIA. 75 

simple than equation (4) of No. (497) ; but it belongs to the parabola 
alone, whereas the equation (4) equally regards every species of conic 
section. 

501. We perceive by the foregoing investigation, that the determina 
tion of the parabolic orbits of the comets, leads to more equations than 
unknown quantities ; and that, therefore, in combining these equations in 
different ways, we can form as many different methods of calculating the 
orbits. Let us examine those which appear to give the most exact re 
sults, or which seem least susceptible of the errors of observations. 

It is principally upon the values of the second differences f-r ^] and 

/d 2 d\ 

( -j ; ), that these errors have a sensible influence. In fact, to determine 

\d. t~s 

them, we must take the finite differences of the geocentric longitudes and 
latitudes of the comet, observed during a short interval of time. But 
these differences being less than the first differences, the errors of obser 
vations are a greater aliquot part of them ; besides this, the formulas of 
No. 495 which determine, by the comparison of observations, the values 

c , /d\ /d 0\ /d 2 \ , /d 2 d\ . . , . . . 

ot , 6, Ijriji (TT) ITTTJ anc ^ VTTV S we greater precision the 

four first of these quantities than the two last. It is, therefore, desirable 
to rest as little as possible upon the second differences of and 6; and 
since we cannot reject both of them together, the method which employs 
the greater, ought to give the more accurate results. This being granted 
let us resume the equations found in Nos. 497, &c. 

* = dr 2 i + 2Rrc St(A ~ a) " f RJ; 



x R sin. (A ) fj_ H_ 
J : /cUv IK> " g 3 /" 

2 Iff*! 



Y sin.* cos. A 
J ( 

~7d ^ i 
vai> 



R sin. 5 cos. ^ cos. (A a) 



76 A COMMENTARY ON [SECT. XL 



-! co, _ 



+ 2 , (**) {(R - 1) sin. (A - ) + "MA )} 



i 
h If 5 



/(I 2 $\ 

If we wish to reject (^ 5) , we consider only the first, second and fourth 

of those equations. Eliminating (-7 ) from the last by means of the 

second, we shall form an equation which cleared of fractions, will contain 
a term multiplied by g 6 r 2 , and other terms affected with even and odd 
powers of r and g. If we put into one side of the equation all the terms 
affected with even powers of g, and into the other all those which involve 
its odd powers, and square both sides, in order to have none but even 
powers of f, the term multiplied by 6 r 2 will produce one multiplied by 
g 12 r 4 . Substituting, therefore, instead of g 2 , its value given by the first 
of equations (L), we shall have a final equation of the sixteenth degree in 
r. But instead of forming this equation in order afterwards to resolve it, 
it will be more simple to satisfy by trial the three preceding ones. 

If we wish to reject ( -, 5), we must consider the first, third and fourth 

of equations (L). These three equations conduct us also to a final equa 
tion of the sixteenth degree in r ; and we can easily satisfy by trial. 

The two preceding methods appear to be the most exact, which we can 
employ in the determination of the parabolic orbits of the comets. It is 
at the same time necessary to have recourse to them, if the motion of the 
comet in longitude or latitude is insensible, or too small for the errors of 
observations sensibly to alter its second difference. In this case, we must 
reject that of the equations (L), which contains this difference. But al 
though in these methods, we employ only three equations, yet the fourth 
is useful to determine amongst all the real and positive values of r, which 
satisfy the system of three equations, that which ought to be selected. 

502. The elements of the orbit of a comet, determined by the above 
process, would be exact, if the values of a, 6 and their first and second 
differences, were rigorous ; for we have regarded, after a very simple 
manner, the excentricity of the terrestrial orbit, by means of the radius- 
vector R of the earth, corresponding to its true anomaly + & right an 
gle ; we are therefore permitted only to neglect the square of this excen- 



BOOK I.] NEWTON S PIUNCIP1A. 77 

tricity, as too small a fraction to produce by its omission a sensible influ 
ence upon the results. But 0, a and their differences, are always suscep 
tible of any degree of inaccuracy, both because of the errors of observa 
tions, and because these differences are only obtained approximately. It 
is therefore necessary to correct the elements, by means of three distant 
observations, which can be done in many ways ; for if we know nearly, 
two quantities relative to the motion of a comet, such that the radius-vec 
tor corresponding to two observations, or the position of the node, and 
the inclination of the orbit ; calculating the observations, first with these 
quantities and afterwards with others differing but little from them, the 
law of the differences between the results, will easily show the necessary 
corrections. But amongst the combinations taken two and two, of the 
quantities relative to the motion of comets, there is one which ought to 
produce greatest simplicity, and which for that reason should be selected. 
It is of importance, in fact, in a problem so intricate, and complicated, to 
spare the calculator all superfluous operations. The two elements which 
appear to present this advantage, are the perihelion distance, and the 
instant when the comet passes this point. They are not only easy to be 



derived from the values of r and -p- ) ; but it is very easy to correct them 

by observations, without being obliged for every variation which they 
undergo, to determine the other corresponding elements of the orbit. 
Resuming the equation found in No. 492 



a (1 e 2 ) is the semi-parameter of the conic section of which a is the 
semi axis-major, and a e the excentricity. In the parabola, where a is 
infinite, and e equal to unity, a (1 e 2 ) is double the perihelion dis 
tance : let D be this distance : the preceding equation becomes relatively 
to this curve 



pde d P 2 r 2 

-. is equal to-^ 5 -; in substituting for e 2 its value r-:+2RrX 

cl t at 2 cos. 2 



COS. 

R> 



cos. (A a) + R 2 , and for (-3-7] and (^rr) 1 their values found in 
No. 499, we shall have 



d t cos. 2 & 



78 A COMMENTARY ON [SECT. XI. 

+ r{(R _ 1) cos. (A-)-. gin -<^- g) } 



+ r R ~ sin. (A ) + R (R 1). 

Let P represent this quantity ; if it is negative, the radius-vector de 
creases, and consequently, the comet tends towards its perihelion. But 
it goes off into the distance, if P is negative. We have then 

D = S -IP* ; 

the angular distance v of the comet from its perihelion, will be determined 
from the polar equation to the parabola, 

cor - 2 ! v = 7 ; 

and finally we shall have the time employed to describe the angle v, by 
the table of the motion of the comets. This time added to or subtracted 
from that of the epoch, according as P is negative or positive, will give 
the instant when the comet passes its perihelion. 

503. Recapitulating these different results, we shall have the following 
method to determine the parabolic orbits of the comets. 

General method of determining the orbits of the comets. 

This method will be divided into two parts ; in the first, we shall give 
the means of obtaining approximately, the perihelion distance of the comet 
and the instant of its passage over the perihelion ; in the second, we shall 
determine all the elements of the orbit on the supposition that the former 
are known. 

Approximate determination of the Perihelion distance of the comet, and 
the instant of its passage over the perihelion, 

We shall select three, four, five, &c. observations of the comet 
equally distant from one another as nearly as possible ; with four obser 
vations we shall be able to consider an interval of 30 ; with five, an in 
terval of 36, or 40 and so on for the rest ; but to diminish the in 
fluence of their errors, the interval comprised between the observations 
must be greater, in proportion as their number is greater. This being 
supposed, 

Let /3, /3 , (3", &c. be the successive geocentric longitudes of the comet, 
7, /, / the corresponding latitudes, these latitudes being supposed positive 
or negative according as they are north or south. We shall divide the dif 
ference 13 |8, by the number of days between the first and second ob 
servation ; we shall divide in like manner the difference ft" P by the 



BOOK I.] NEWTON S PRINCIPIA. 79 

number of days between the second and third observation ; and so on. 
Let 3 8, d B , d B", &c. be these quotients. 

We next divide the diffei-ence 88 SB by the number of days be 
tween the first observation and the third ; we divide, in like manner, the 
difference 8 B" d $ by the number of days between the second and 
fourth observations ; similarly we divide the difference 8 B" 8 B" by the 
number of days between the third and fifth observation, and so on. Let 
8 2 18, 8 2 6 , & 2 /3", &c. denote these quotients. 

Again, we divide the difference B z B 8 2 B by the number of days 
which separate the first observation from the fourth ; we divide in like 
manner 8 2 B" 8 2 B by the number of days between the second obser 
vation and the fifth, and so on. Make 8 3 8, 8 3 8 , &c. these quotients. 
Thus proceeding, we shall arrive at 8 n - l 8 9 n being the number of obser 
vations employed. 

This being done, we proceed to take as near as may be a mean epoch 
between the instants of the two extreme observations, and calling i, i , i", 
&c. the number of days, distant from each observation, i, i , i", Sec. ought 
to be supposed negative for the observations made prior to this epoch ; 
the longitude of the comet, after a small number z of days reckoned from 
the Epoch will be expressed by the following formula : 

j3 _ i a 8 + i i d 2 B i i i" 8 3 B + &c. 
\ +Z J3 8(i + i )8 Z 8+ (i i + i i"+i i")3 3 B (i i i"+i i i " + i i" i "+. . (p) 

)i i"i" 

V. 2 



The coefficients of 8 8, + 8 z B, 8 3 8, &c. in the part independent 
of z are 1st the numbers i and i , secondly the sum of the products two 
and two of the three numbers i, i , \" ; thirdly the sum of the products 
three and three, of the four numbers i, i , i", i" , &c. 

The coefficients of 8 3 B, + 8 4 8, 8 5 8, &c. in the part multiplied 
by z 2 , are first, the sum of the three numbers i, i , i 7 ; secondly of the 
products two and two of the four numbers i, i , i , i "; thirdly the sum of 
the products three and three of the five numbers i, i , i", i" , i"", &c. 

Instead of forming these products, it is as simple to develope the func 
tion B + (z i) 6/3 + (z i) (z i ) 6 2 /3+ (z i) (z i ) (z i") 
X 6 3 8 -f- &c. rejecting the powers of z superior to the square. This 
gives the preceding formula. 

If we operate in a similar manner upon the observed geocentric lati 
tudes of the comet ; its geocentric latitude, after the number z of days 
from the epoch, will be expressed by the formula (p) in changing 8 into 
7. Call (q) the equation (p) thus altered. This being done, 



80 A COMMENTARY ON [SECT. XL 

a will be the part independent of z in the formula (p) ; and 6 that in the 
formula (q). 

Reducing into seconds the coefficient of z in the formula (p), and 
taking from the tabular logarithm of this number of seconds, the logarithm 
4,0394622, we shall have the logarithm of a number which we shall de 
note by a. 

Reducing into seconds the coefficients of z 2 in the same formula, and tak- 
ing from the logarithm of this number of seconds, the logarithm 1.9740144, 
we shall have the logarithm of a number, which we shall denote by b. 

Reducing in like manner into seconds the coefficients of z and z 2 in 
the formula (q) and taking away respectively from the logarithms of these 
numbers of seconds, the logarithms, 4,0394622 and 1,9740144, we shall 
have the logarithms of two numbers, which we shall name h and 1. 

Upon the accuracy of the values of a, b, h, 1, depends that of the 
method; and since their formation is very simple, we must select and 
multiply observations, so as to obtain them with all the exactness which 
the observations will admit of. It is perceptible that these values are only 

/dax /d 2 ax /d 6\ /d 2 6\ 
the quantities (^J > VdT 2 / \d~t/ VdT 2 / wmch we have ex P ress 

ed more simply by the above letters. 

If the number of observations is odd, we can fix the Epoch at the 
instant of the mean observation; which will dispense with calculating the 
parts independent of z in the two preceding formulas ; for it is evident, 
that then these parts are respectively equal to the longitude and latitude 
of the mean observation. 

Having thus determined the values of a, a, b, 8, h, and 1, we shall de 
termine the longitude of the sun, at the instant we have selected for the 
epoch, R the corresponding distance of the Earth from the sun, and R 
the distance which answers to E augmented by a right angle. We shall 
have the following equations 

(1) 

(2) 

( 



! 

y 
y 


X 2 

cos. 2 6 
^ sin. (E a 


x cos. (E a) + R 2 

) r i i i bx 






2a 
= x -j h tan. d 


( g 3 R 3 j 2 a 

1 l a 2 sin. 6 . cos. 


iVi 


1 2h 2h 


if 



R sin. ^ cos. ^ f ^ ^ (3; 

-tr~ cos - 



BOOK I.] NEWTON S PRINCIPIA. 81 

(IT 1) cos. (E a)} 2 a x -[(R/ 1) sin. (E a) + 





To derive from these equations the values of the unknown quantities 
Xj y> & we must consider, signs being neglected, whether b is greater or 
less than 1. In the first case we shall make use of equation (1), (2), and 
(4). We shall form a first hypothesis for x, supposing it for instance 
equal to unity; and we then derive by means of equations (1), (2), the 
values of and of y. Next we substitute these values in the equation (4) ; 
and if the result is 0, this will be a proof that the value of x has been 
rightly chosen. But if it be negative we must augment the value of x, 
and diminish it if the contrary. We shall thus obtain, by means of a 
small number of trials the values of x, y and g. But since these unknown 
quantities may be susceptible of many real and positive values, we must 
seek that which satisfies exactly or nearly so the equation (3). 

In the second case, that is to say, if 1 be greater than b, we shall use 
the equations (1), (3), (4), and then equation (2) will give the verifi 
cation. 

Having thus the values of x, y, g, we shall have the quantity 

p = & + h x tan> 6} ~ R y cos - (E K) 



+ x = _(R _ 1) cos . (E *) Rax rin (E-) 

+ R.(R/ 1). 
The Perihelion distance D of the comet will be 

D = s -lp* ; 

the cosine of its anomaly v will be given by the equation 

1 D 

cos^-v = -; 

and hence we obtain, by the table of the motion of the comets, the time 
employed to describe the angle v. To obtain the instant when the comet 
passes the perihelion, we must add this time to, or subtract it from the 
epoch according as P is negative or positive. For in the first case the 
comet approaches, and in the second recedes from, the perihelion. 

Having thus nearly obtained the perihelion distance of the comet, and 
the instant of its passage over the perihelion ; we are enabled to correct 
them by the following method, which has the advantage of being inde 
pendent of the approximate values of the other elements of the orbit. 

Vot. IT. F 



82 A COMMENTARY ON [SECT. XI. 

An exact Determination of the elements of the orbit, when we know ap 
proximate values of the perihelion distance of the comet, and of the instant 
of its passage over the perihelion. 

We shall first select three distant observations of the comet; then 
taking the perihelion distance of the comet, and the instant of its crossing 
the perihelion, determined as above, we shall calculate the three anomalies 
of the comet and the corresponding radius-vectors corresponding to the 
instants of the three observations. Let v, v , v" be these anomalies, those 
which precede the passage over the perihelion being supposed negative. 
Also let g, g g" be the corresponding radius-vectors of the comet ; then 
v 7 v, V v will be the angles comprised by g and g and by , g". 
Let U be the first of these angles, U the second. Again, call a, a a! the 
three observed geocentric longitudes of the comet, referred to a fixed 
equinox ; 6, 6 , 6" its three geocentric latitudes, the south latitudes being 
negative. Let ft, ft , ft be the three corresponding heliocentric longi 
tudes and *r, w t &") its three heliocentric latitudes. Lastly call E, E , E" 
the three corresponding longitudes of the sun, and R, R , R" its three 
distances to the center of the earth. 

Conceive that the letter S indicates the center of the sun, T that of the 
earth, and C that of the comet, C that of its projection upon the plane 
of the ecliptic. The angle S T C is the difference of the geocentric lon 
gitudes of the sun and of the comet. Adding the logarithm of the cosine 
of this angle, to the logarithm of the cosine of the geocentric latitude of 
the comet, we shall have the logarithm of the cosine of the angle S T C. 
We know, therefore, in the triangle S T C, the side S T or R, the side 
S C or g, and the angle S T C, to find the angle C S T. Next we shall 
have the heliocentric latitude -a of the comet, by means of the equation 

sin. 6 sin. C S T 



sin. =r 



sin. C T S 

The angle T S C is the side of a spherical right angled triangle, of 
which the hypothenuse is the angle T S C, and of which one of the sides 
is the angle . Whence we shall easily derive the angle T S C 7 , and con 
sequently the heliocentric longitude ft of the comer. 

We shall have after the same manner ~ , ; ", ft" ; and the values of 
ft, ft , ft" will show whether the motion of the comet be direct or retro 
grade. 

If we imagine (he two arcs of latitude -, t/, to meet at the pole of the 
ecliptic, they would make there an angle equal to ft ft; and in the 



BOOK I.] NEWTON S PRINCIPIA. 83 

spherical triangle formed by this angle, and by the sides w, - -a 

it being the semi-circumference, the side opposite to the angle (3 f (3 
will be the angle at the sun comprised between the radius-vectors g, and 
f . We shall easily determine this by "spherical Trigonometry, or by the 
formula 

sin. 2 i V = cos. 2 ^ ( w + ** ) cos 2 - - (& 0) cos. cos. , 
& & 

in which V represents this angle ; so that if we call A the angle of which 
the sine squared is 

cos 2 - (ft ]8) cos. 9 . cos. , 

i) 

and which we shall easily find by the tables, we shall have 
- ^. i V = cos. (i . + I , + A) cos. ( \, + i ra _ A ). 

If in like manner we call V the angle formed by the two radius-vectors 
S) ?"} w e have 

sin.iv=cos.(l.+ ^ + 

A being what A becomes, when , /3 are changed into w", /3". 

If, however, the perihelion distance and the instant of the comet s 
crossing the perihelion, were exactly determined, and if the observations 
were rigorously exact, we should have 

V = U, V = U ; 

But since that is hardly ever the case, we shall suppose 
m = U V ; m = U V. 

We shall here observe that the revolution of the triangle S T C, gives 
for the angle C S T two different values : for the most part the nature 
of the motion of the comets, will show that which we ought to use, and 
the more plainly if the two values are very different ; for then the one will 
place the comet more distant from the earth, than the other, and it will 
be easy to judge, by the apparent motion of the comet at the instant of 
observation, which ought to be preferred. But if there remains any un 
certainty, we can always remove it, by selecting the value which renders 
V and V least different from U and U . 

We next make a second hypothesis in which, retaining the same pas 
snge over the perihelion as before, we shall suppose the perihelion dis 
tance to vary by a small quantity ; for instance, by the fiftieth part of 

F2 



84 A COMMENTARY ON [SECT. XL 

its value, and we shall investigate on this hypothesis, the values of U V, 
U \ T/ . Let then 

n = U V ; n = U V. 

Lastly, we shall frame a third hypothesis, in which, retaining the same 
perihelion distance as in the first, we shall suppose the instant of the pas 
sage over the perihelion to vary by a half-day, or a day more or less. In 
this new hypothesis we must find the values of 
U VandofU V; 
which suppose to be 

p = U - V, p = U V. 

Again, if we suppose u the number by -which we ought to multiply the 
supposed variation in the perihelion distance in order to make it the 
true one, and t the number by which we ought to multiply the supposed 
variation of the instant when the comet passes over the perihelion in 
order to make it the true instant, we shall have the two following equa 
tions : 

(m n ) u + (m p ) t = m ; 
(m n ) u + (m p ) t = m ; 

whence we derive u and t and consequently the perihelion distance cor 
rected, and the true instant of the comet s passing its perihelion. 

The preceding corrections suppose the elements determined by the 
first approximation, to be sufficiently near the truth for their errors to be 
regarded as infinitely small. But if the second approximation should 
not even suffice, we can have recourse to a third, by operating upon the ele 
ments already corrected as we did upon the first ; provided care be taken to 
make them undergo smaller variations. It will also be sufficient to calculate 
by these corrected elements the values of U V, and of U V. Call 
ing them M, M , we shall substitute them for m, m in the second mem 
bers of the two preceding equations. We shall thus have two new equa 
tions which will give the values of u and t, relative to the corrections of 
these new elements. 

Thus having obtained the true perihelion distance and the true instant 
of the comet s passing its perihelion, we obtain the other elements of the 
orbit in this manner. 

Let j be the longitude of the node which would be ascending if the 
motion of the comet were direct, and <p the inclination of the orbit. We 
shall have by comparison of the first and last observation, 

tan, -a sin. /3 tan. */ sin. /3 ^ 





tan< - 



tan. cos. jS" tan. " cos. 



BOOK I.] NEWTON S PRINCIPIA. 85 

tan. -a" 

tan. = . - 7-577 - rr . 

sin. (/3"_ j) 

Since we can compare thus two and two together, the three observa 
tions, it will be more correct to select those which give to the above frac 
tions, the greatest numerators and the greatest denominators. 

Since tan. j may equally belong to j and <x + j, j being the smallest of 
the positive angles containing its value, in order to find that which we 
ought to fix upon, we shall observe that <p is positive and less than a right 
angle ; and that sin. (/3" j) ought to have the same sign as tan. -a". 
This condition will determine the angle j, and this will be the position 
of the ascending node, if the motion of the comet is direct ; but if retro 
grade we must add two right angles to the angle j to get the position of 
the node. 

The hypothenuse of the spherical triangle whose sides are $" j and 
w", is the distance of the comet from its ascending node in the third ob 
servation; and the difference between v" and this hypothenuse is the 
interval between the node and the perihelion computed along the orbit. 

If we wish to give to the theory of a comet all the precision which ob 
servations will admit of, we must establish it upon an aggregate of the best 
observations ; which may be thus done. Mark with one, two, &c. dashes 
or strokes the letters m, n, p relative to the second observation, the third, 
&c. all being compared with the first observation. Hence we shaH form 
the equations 

(m n ) u + (m p ) t = m 
(m n ) u + (m 7 p ) t = m 
(m" n") u + (m" p") t = m" 

&c. = &c. 

Again, combining these equations so as to make it easier to determine 
u and t, we shall have the corrections of the perihelion distance and of the 
instant of the comet s passing its perihelion, founded upon the aggregate 
of these observations. We shall have the values of 

ft , 8", &C. , , w", &C., 

and obtain 

. __ tan. * (sin. 3 + sin. B" + &c.) sin. |8 (tan. / + tan. " + &c.) 
" J ~~ tan. * (cos. B + cos. B" + &c.) cos. B (tan. + tan. " + &c.) 

_ tan. tar -f- tan. r" + &c. 

* ~ 



sin. (/3 j) + sin. (B" j) + &c. 
504. There is a case, very rare indeed, in which the orbit of a comet 
can be determined rigorously and simply ; it is that where the comet has 
been observed in its two nodes. The straight line which joins these 

F3 



86 A COMMENTARY ON [SECT* XI. 

two observed positions, passes through the center of the sun and coincides 
with the line of the nodes. The length of this straight line is determined 
by the time elapsed between the two observations. Calling T this time 
reduced into decimals of a day, and denoting by c the straight line in 
question, we shall have (No. 493) 

3 

1 / T 2 

= 2 */ ( 9.688724) 2 * 

Let /3 be the heliocentric longitude of the comet, at the moment of the 
first observation ; f its radius- vector ; r its distance from the earth ; and a 
its geocentric longitude. Let, moreover, R be the radius of the terrestrial 
orbit, at the same instant, and E the corresponding longitude of the sun. 
Then we shall have 

g sin. (3 = r sin. a R sin. E ; 
g cos. (3 = r cos. a R cos. E. 

Now cr + j3 will be the heliocentric longitude of the comet at the in 
stant of the second observation ; and if we distinguish the quantities g, , 
r, R, and E relative to this instant by a dash, we shall have 
o sin. B = R sin. E r sin. a ; 
g cos. 3 = R cos. E r cos. a . 
These four equations give 

_ r sin R sin. E _ r sin.a R sin. E 
tan< " ~ rcos.a Rcos.E r cos. a R cos. E 
whence we obtain 

, _ R R sin. (E E ) R r sin. ( E ) 

r sin. (a a) R sin. (a! E) 
We have also 

(g 4. ) sin. /3 = r sin. a r sin. a R sin. E + R sin. E 
(g -j_ g ) cos. j8 = r cos. a r cos. a R cos. E + R 7 cos. E 7 . 
Squaring these two equations, and adding them together, and substitut 
ing c for g + g , we shall have 

c 2 = R 2 2RR cos.(E E) + R /2 
+ 2 r {R cos. ( EO R cos. (a E)} 
+ 2 r {R cos. (a E) R cos. (a E )l 
+ r 2 2rr cos. (a a) + r /2 . 

If we substitute in this equation for r its preceding value in terms of r, 
we shall have an equation in r of the fourth degree, which can be resolved 
by the usual methods. But it will be more simple to find values of r, r 
by trial such as will satisfy the equation. A few trials will suffice for that 
purpose. 



BOOK I.] NEWTON S PRINCIPIA. 87 

By means of these quantities we shall have /3, g and g . If v be the 
angle which the radius g makes with the perihelion distance called D ; 
<r v will be the angle formed by this same distance, and by the radius g . 
We shall thus have by the equation to the parabola 

D D 

S = 



1 s 1 

cos. 2 v sin. 2 v 

; * rw 



which give 



o 



. . 

2 r g + s 

We shall therefore have the anomaly v of the comet, at the instant of 
the first observation, and its perihelion distance D, whence it is easy to 
find the position of the perihelion, at the instant of the passage of the 
comet over that point. Thus, of the five elements of the orbit of the co 
met, four are known, namely, the perihelion distance, the position of the 
perihelion, the instant of the comet s passing the perihelion, and the posi 
tion of the node. It remains to learn the inclination of the orbit; but for 
that purpose it will be necessary to have recourse to a third observation, 
which will also serve to select from amongst the real and positive roots of 
the equation in r, that which we ought to make use of. 

505. The supposition of the parabolic motion of comets is not rigorous ; 
it is, at the same time, not at all probable, since compared with the cases 
that give the parabolic motion, there is an infinity of those which give the 
elliptic or hyperbolic motions. Besides, a comet moving in either a para 
bolic or hyperbolic orbit, will only once be visible; thus we may with 
reason suppose these bodies, if ever they existed, long since to have dis 
appeared ; so that we shall now observe those only which, moving in or 
bits returning into themselves, shall, after greater or less incursions into 
the regions of space, again approach their center the sun. By the follow 
ing method, we shall be able to determine, within a few years, the period 
of their revolutions, when we have given a great number of very exact 
observations, made before and after the passage over the perihelion. 

Let us suppose we have four or a greater number of good observations, 
which embrace all the visible part of the orbit, and that we have deter 
mined, by the preceding method, the parabola, which nearly satisfies these 
observations. Let v, v , v", v ", &c. be the corresponding anomalies; 
1 & t "9 f" > & c - tne radius-vectors. Let also 

v v = U, v" v = U , v" v = U", &c. 



88 A COMMENTARY ON [SECT. XL 

Then we shall estimate, by the preceding method with the parabola 
already found, the values of U, U , U", &c., V, V, V", &c. Make 
m = U V, m = U V 7 , m" = U" V", &c. 

Next, let the perihelion distance in this parabola vary by a very small 
quantity, and on this hypothesis suppose 

n = U V; n = U V; n" = U" V", &c. 
We will form a third hypothesis, in which the perihelion distance re 
maining the same as in the first, we shall make the instant of the comet s 
passing its perihelion vary by a very small quantity ; in this case let 

p = U V; p = U V; p" = U" V"; &c. 
Lastly, we shall calculate the angle v and radius g, with the perihelion 
distance, and instant over the perihelion on the first hypothesis, supposing 
the orbit an ellipse, and the difference 1 e between its excentricity and 
unity a very small quantity, for instance JQ. To get the angle v, in this 
hypothesis, it will suffice (489) to add to the anomaly v, calculated in the 
parabola of the first hypothesis, a small angle whose sine is 

-TJ. (1 e) tan. v -| 4 3 cos. 2 v 6 cos. 4 v f . 
J.U \ & y 

Substituting afterwards in the equation 

D 

s = 

cos 2 1 

for v, this anomaly, as calculated in the ellipse, we shall have the corre 
sponding radius-vector g. After the same manner, we shall obtain v , g t 
v", ", &c. Whence we shall derive the values of U, U , U", &c. and 
(by 503) of V, V, V", &c. 
In this case let 

q = U V; q = U V , q" = U" V", &c. 
Finally, call u the number by which we ought to multiply the supposed 
variation in the perihelion distance, to make it the true one ; t the number 
by which we ought to multiply the supposed variation in the instant over 
the perihelion, to make it the true instant ; and s that by which we should 
multiply the supposed value of 1 e, in order to get the true one ; and 
we shall obtain these equations : 

( m n) u + (m p) t + (in q; s = m; 
(m n ) u + (m p ) t + (m q ) s = m ; 
(m" n") u + (m" p") t + (m" q") s = m" ; 
(m " _ n " ) u + (m " p ") t + (m" q ") s = m "; 

&c. 



BOOK 1.] NEWTON S PRINCIPIA. 89 

We shall determine, by means of these equations, the values of u, t, s; 
whence will be derived the true perihelion distance, the true instant over 
the perihelion, and the true value of 1 e. Let D be the perihelion 
distance, and a the semi-axis major of the orbit; then we shall have 

a = - ; the time of a sidereal revolution of the comet, will be expressed 
1 e 

2. / f) 5. 

by a number of sidereal years equal to a 2 or to (- -)*> tne mean 

\ J^ __ Q/ 

distance of the sun from the earth being unity. We shall then have 
(by 503) the inclination of the orbit and the position of the node. 

Whatever accuracy we may attribute to the observations, they will 
always leave us in uncertainty as to the periodic times of the comets. To 
determine this, the most exact method is that of comparing the observa 
tions of a comet in two consecutive revolutions. But this is practicable, 
only when the lapse of time shall bring the comet back towards its peri 
helion. 

Thus much for the motions of the planets and comets as caused by the 
action of the principal body of the system. We now come to 

506. General methods of determining by successive approximations, the 
motions of the heavenly bodies. 

In the preceding researches we have merely dwelt upon the elliptic 
motion of the heavenly bodies, but in what follows we shall estimate them 
as deranged by perturbing forces. The action of these forces requires only 
to be added to the differential equations of elliptic motion, whose integrals 
in finite terms we have already given, certain small terms. We must deter 
mine, however, by successive approximations, the integrals of these same 
equations when thus augmented. For this purpose here is a general me 
thod, let the number and degree of the equations be what they may. 

Suppose that we have between die n_ variables y, y , y", &c. and the 
time t whose element d t is constant, the n differential equations 



= 



&c. = &c. 

P, Q, P , Q , &c. being functions of t, y, y , &c. and of the differences to 
the order i 1 inclusively, and a being a very small constant coefficient, 
which, in the theory of celestial motions, is of the order of the perturb 
ing forces. Then let us suppose we have the finite integrals of those 



90 A COMMENTARY ON [SECT. XL 

equations when Q, Q , &c. are nothing. Differentiating each i 1 
times successively, we shall form with their differentials i n equations by 
means of which we shall determine by elimination, the arbitrary constants 
c, c , c", &c. in functions of t, y, y , y", &c. and of their differences to the 
order i 1. Designating therefore by V, V, V", &c. these functions 
we shall have 

c = V; c = V; c" = V"; &c. 

These equations are the i n integrals of the (i l) th order, which the 
equations ought to have, and which, by the elimination of the differences 
of the variables, give their finite integrals. 

But if we differentiate the preceding integrals of the order i 1, we 
shall have 

= dV; = d V; = d M" ; &c. 

and it is clear that these last equations being differentials of the order i 
without arbitrary constants, they can only be the sums of the equations 



= &c. 

each multiplied by proper factors, in order to make these sums exact dif 
ferences. Calling, therefore, F d t, F d t , &c. the factors which ought 
respectively to multiply them in order to make = d V ; also in like 
manner making H d t, H d t , &c. the factors which would make On d V, 
and so on for the rest, we shall have 



&c. 

F, F , &c. H, H 7 , &c. are functions of t, y, y , y", &c. and of their dif 
ferences to the order i 1. It is easy to determine them when V, V 7 , &c. 

are known. For F is evidently the coefficient of -r- ^ in the differential 
of V ; F is the coefficient of ^ in the same differential, and so on. 

Cl L 

d v d y 
In like manner, H, H , &c. are the coefficients of T -f , -,- j > &c. in the 

Cl t Cl L 

differential of V 7 . Thus, since we may suppose V, V , &c. known, by dif- 



BOOK I.] NEWTON S PRINCIPIA. 91 

(I * " * v Cl V 

ferentiating with regard to , . _\ , , i _ l , &c. we shall have the 
factors by which we ought to multiply the differential equations 
= |if + P, = ^i + P , &c. 

in order to make them exact differences. 
Now resume the differential equations 

= ^-"-f + P + . Q ; = -^- y r -f- F + a . Q , &c. 

If we multiply the first by F d t, the second by F d t, and so on, we 
shall have by adding the results 

= d V + a d t {F Q + F Q + &c.}, 
In the same manner, we shall have 

= d V + a d t JH Q + H Q + &c.} 
&c. 
whence by integration 

c _ a/d t {F Q + F Q + &c.} = V; 
c _ a/d t {H Q + H Q + &c.J = V; 
&c. 

We shall thus have z n differential equations, which will be of the same 
form as in the case when Q, Q , &c. are nothing, with this only differ 
ence, that the arbitrary constants c, c , c", &c. must be changed into 

c_a/dt {FQ+FQ / +&c.}, c a/dtfHQ+ H Q +&c.}&c. 
But if. in the supposition of Q, Q , &c. being equal to zero, we eliminate 
from the z n integrals of the order i 1, the differences of the variables 
y, y , &c. we shall have n finite integrals of the proposed equations. We 
shall therefore have these same integrals when Q, Q , &c. are not zero, by 
changing in the first integrals, c, c , &c. into 

c a /d t FQ + &c.}, c /d t {H Q -f- &c.}&c. 
507. If the differentials 

d t F Q + F Q + &c.J, d t {II Q + H Q + &c.J&c. 
are exact, we shall have, by the preceding method, finite integrals of the 
proposed differentials. But this is not so, except in some particular cases, 
of which the most extensive and interesting is that in which they are 
linear. Thus let P, P , &c. be linear functions of y, y , &c. and of their 
differences up to the order i 1, without any term independent of these 
variables, and let us first consider the case in which Q, Q , &c. are no 
thing. The differential equations being linear, their successive integrals 



92 A COMMENTARY ON [SECT. XL 

are likewise linear, so that c = V, c = V, &c. being the i n integrals of 
the order i ], of the linear differential equations 



V, V, &c. may be supposed linear functions of y y , &c. and of their dif 
ferences to the order i 1. To make this evident, suppose that in the 
expressions for y, y , &c. the arbitrary constant c is equal to a determinate 
quantity plus an indeterminate d c ; the arbitrary constant c equal to a 
determinate quantity plus an indeterminate 5 c &c. ; then reducing these 
expressions according to the powers and products of d c, d c , &c. we shall 
have by the formulas of No. 487 



1.2 






2 

&c. 

Y, Y , f~j J , &c. being functions oft without arbitrary constants. Sub 

stituting those values, in the proposed differential equations, it is evident . 
that d c, d c , &c. being indeterminate, the coefficients of the first powers 
of such of them ought to be nothing in the several equations. But these 
equations being linear, we shall evidently have the terms affected with the 
first powers of 8 c, d c , &c. by substituting for y, y , &c. these quantities 
respectively 



These expressions of y, y , &c. satisfy therefore separately the proposed 
equations ; and since they contain the i n arbitraries d c, d c , &c. they are 
complete integrals. Thus we perceive, that the arbitraries are under a 
linear form in the expressions of y, y , &c. and consequently also in their 
differentials. Whence it is easy to conclude that the variables y, y , &c. 
and their differences, may be supposed to be linear in the successive inte 
grals of the proposed differential equations. 

d v d * v 
Hence it follows, that F, F , &c. being the coefficients of y r- , - J . , 

Cl t Cl t 



BOOK L] NEWTON S PRINCIPLE 93 

&c. in the differential of V ; H, H , &c. being the coefficients of the same 
differences in the differentia] of V, &c. these quantities are functions of 
variable t only. Therefore, if we suppose Q, Q , &c. functions of t alone, 
the differentials 

d t {F Q + F Q + &c.] ; d t [B Q + IF Q + &c.$ ; &c. 
will be exact. 

Hence there results a simple means of obtaining the integrals of any 
number whatever n of linear differential equations of the order i, and 
which contain any terms a Q, a Q , &c. functions of one variable t, having 
known the integrals of the same equations in the case where Q, Q 7 , &c. 
are supposed nothing. For then if we differentiate their n finite integrals 
i 1 times successively, we shall have i n equations which will give, by 
elimination, the values of the i n arbitrary constants c, c , &c. in functions 
of t, y, y , &c. and of their differences to the i 1 th order. We shall thus 
form the i n equations c = V, c = V, &c. This being done, F, F , &c. 

( ] i - 1 y (J i - 1 y 

will be the coefficients of -r j~ , Trntrs & Ct * n ^ ^ ^ /J & c * w ^ 

be the coefficients of the same differences in V, and so on. We shall, 
therefore, have the finite integrals of the linear differential equations 

o = + P + Q; o = + p + Q ; &c. 



by changing, in the finite integrals of these equations deprived of their last 
terms a Q, a Q , &c. the arbitrary constants c, c , &c. into 

c /d t F Q + F Q +&c.k c a/d t {U Q + H Q +&c.| &c. 
Let us take, for example, the linear equation 



The finite integral of the equation 



is (found by multiplying by cos. a t, and then by parts getting 
f cos. a t . *}- = cos. a t ~- + a f sin. a t , j- . d t = cos. a t . -~ + 

a sin. a t . y a 2 f cos. a t . y . . c = a cos. a t . -*-; + a sin. a t . y, &c.) 



c c 

y = sin. a t + cos. a t, 
a a 



c, c being arbitrary constants. 



94 A COMMENTARY ON [SECT. XI. 

This integral gives by differentiation 

dy 

-r* = c cos. at c sin. a t. 
d t 

If we combine this with the integral itself, we shall form two integrals 
of the first order 

d v 
c = a y sin. a t + -r-i cos. a t ; 





c = a y cos. at -- r-^- sin. a t ; 

and therefore shall have in this case 

F = cos. at; H = sin. a t, 
and the complete integral of the proposed equation will therefore be 

c c cc sin. a t _ . 

y = sm. a t -4 -- cos. at -- / U d t cos. a t 
a a a J 

a, cos. a t rf ^ , . 
-\ -- j Q d t sin. a t. 

Hence it is easy to conclude that if Q is composed of terms of the form 

K . (m t -4- i) each of these terms will produce in the value of y the 
cos. v * 

corresponding term 

K sin. . 

2 ---- ,. (m t + e). 
m 2 a 2 cos. v 

If m be equal to a, the term K (m t + t) will produce in y, 1st. the 

term -. - . (a t + ) which being comprised by the two terms 
4 a * cos. v 

c c cc 1C t cos 

sin. a t-\ -- cos. at, may be neglected: 2dly. the term + - . . (a t + g)> 

a a 2 a sm. v 

+ or being used according as the term of Q is a sine or cosine. We 
thus perceive how the arc t produces itself in the values of y, y , &c. with 
out sines and cosines, by successive integrations, although the differentials 
do not contain it in that form. It is evident this will take place when 
ever the functions F Q, F , Q , &c. H Q, H Q , &c. shall contain con 
stant terms. 

508. If the differences 

d t [F Q + &c.}, d t { H Q + &c.} 

are not exact, the preceding analysis will not give their rigorous integrals. 
But it affords a simple process for obtaining them more and more nearly 
by approximation when a is very small, and when we have the values of 



t 



BOOK I.j NEWTON S PRINCIPIA. 95 

y, y 7 , &c. on the supposition of a being zero. Differentiating these values, 
i 1 times successively, we shall form the differential equations of the 
order i 1, viz. 

c = V; c = V, &c. 

d * v d v 
The coefficients of j-4- , -. *- , &c. in the differentials of V, V , &c. 

Cl I *J. L 

being the values of F, F , &c. H, H , Sec. we shall substitute them in the 
differential functions 

d t (F Q + F Q + &c.) ; d t (H Q + H Q + &c) ; &c. 

Then, we shall substitute in these functions, for y, y , &c. their first 
approximate values, which will make these differences functions of t and of 
the arbitrary constants c, c , &c. 

Let T d t, T d t, &c. be these functions. If we change in the first 
approximate values of y, y , &c. the arbitrary constants c, c , &c. re 
spectively into c a y T d t, c a y X d t, &c. we shall have the 
second approximate values of those variables. 

Again substitute these second values in the differential functions 
d t . (F Q + &c.) ; d t (H Q + &c.) &c. 

But it is evident that these functions are then what T d t, T d t, &c. 
become when we change the arbitrary constants c, c , &c. into c af T d t, 
c ufT d t, &c. Let therefore T /5 T/, &c. denote what T, T, &c. 
become by these changes. We shall get the third approximate values of 
y, y , &c. by changing in the first c, c , &c. respectively into c yX, d t, 
c- yX; d t, &c. 

Calling T //} T//, in like manner, what T, X, &c. become when 
we change c, c , &c. into c y T, d t, c y T/ d t, &c. we shall 
have the fourth approximate values of y, y , &c. by changing in the first 
approximate values of these variables into c y T /7 d t, c y X,/ d t, 
&c. and so on. 

We shall see presently that the determination of the celestial motions, 
depends almost always upon differential equations of the form 

= ^y + a y + Q, 

Q being a rational and .integer function of y, of the sine and cosine of 
angles increasing proportionally with the time represented by t. The 
following is the easiest way of integrating this equation. 

First suppose u nothing, and we shall have by the preceding No. a first 
value of y. 

Next substitute this value in Q, which will thus become a rational and 



9G A COMMENTARY ON [SECT. XI. 

entire function of sines and cosines of angles proportional to the time. 
Then integrating the differential equation, we shall have a second value 
of y approximate up to quantities of the order inclusively. 

Again substitute this value in Q, and, integrating the differential equa 
tion, we shall have a third approximation of y, and so on. 

This way of integrating by approximation the differential equations of 
the celestial motions, although the most simple of all, possesses the dis 
advantage of giving in the expressions of the variables y, y , &c. the arcs 
of a circle (symbols sine and cosine] in the very case where these arcs 
do not enter the rigorous values of these variables. We perceive, in 
fact, that if these values contain sines or cosines of angles of the order a t, 
these sines or cosines ought to present themselves in the form of series, in 
the approximate values found by the preceding method ; for these last 
values are ordered according to the powers of . This developement 
into series of the sine and cosine of angles of the order a t, ceases to be 
exact when, by lapse of time, the arc a t becomes considerable. The ap 
proximate values of y, y , &c. cannot extend to the case of an unlimited 
interval of time. It being important to obtain values which include both 
past and future ages, the reversion of arcs of a circle contained by the 
approximate values, into functions which produce them by their develope 
ment into series, is a delicate and interesting problem of analysis. Here 
follows a general and very simple method of solution. 

509. Let us consider the differential equation of the order i, 



d v d *~ * v 

a being very small, and P and Q algebraic functions of y, -^ , . . . . -, j ^ , 

tl L tl L 

and of sines and cosines of angles increasing proportionally with the time. 
Suppose we have the complete integral of this differential, in the case of 
a = 0, and that the value of y given by this integral, does not contain the 
arc t, without the symbols sine and cosine. Also suppose that in inte 
grating this equation by the preceding method of approximation, when a 
is not nothing, we have 

y = X + t Y + t 2 Z + t 3 S + &c. 

X, Y, Z, &c. being periodic functions of t, which* contain the i arbitraries 
c, c , c", &c. and the powers of t in this expression of y, going on to in 
finity by the successive approximations. It is evident the coefficients 
of these powers will decrease with the greater rapidity, the less is a. 
In the theory of the motions of the heavenly bodies, expresses the order 
of perturbing forces, relative to the principal forces which animate them. 



BOOK I.] NEWTON S PRINCIPIA. 97 

d v 
If we substitute the preceding value of y in the function -^H-P-f-aQs 

it will take the form k + k t + k" t 2 + &c., k, k , k", &c. being perio 
dic functions of t ; but by the supposition, the value of y satisfies the dif 
ferential equation 



we ought therefore to have identically 

= k + k t + k" t 2 + &c. 

If k, k , k", &c. be not zero this equation will give by the inversion of 
series, the arc t in functions of sines and cosines of angles proportional to 
the time t. Supposing therefore a to be infinitely small, we shall have t 
equal to a finite function of sines and cosines of similar angles, which is 
impossible. Hence the functions k, k , &c. are identically nothing. 

Again, if the arc t is only raised to the first power under the symbols 
sine and cosine, since that takes place in the theory of celestial motions, 
the arc will not be produced by the successive differences of y. Substi 

tuting, therefore, the preceding value of y, in the function ~ ^+P+ . Q, 

the function of k + k t + &c. to which it transforms, will not contain 
the arc t out of the symbols sine and cosine, inasmuch as it is already con 
tained in y. Thus changing in the expression of y, the arc t, without the 
periodic symbols, into t 0, 6 being any constant whatever, the function 
k + k t + &c. will become k + k (t 6) + &c. and since this last 
function is identically nothing by reason of the identical equations k =r 
k = 0, it results that the expression 

y = X + (t 6} Y + (t 6) 2 Z + &c. 
also satisfies the differential equation 

= ai? + p + Q - 

Although this second value of y seems to contain i + 1 arbitrary con 
stants, namely, the i arbitrages c, c , c", &c. and 6 t yet it can only have i 
distinct ones. It is therefore necessary that by a proper change in the 
constants c, c , &c. the arbitrary 6 be made to disappear, and thus the 
second value of y will coincide with the first. This consideration will fur 
nish us with the means of making disappear the arc of a circle out of the 
periodic symbols. 

Give the following form to the second expression for y : 

y - X + (t - . R. 
V l. II. Cr 



98 A COMMENTARY ON [SECT. XL 

Then supposing 6 to disappear from y, we have 



and consequently 



( - > (irr)- 

Differentiating successively this equation we shall have 



whence it is easy to obtain, by eliminating R and its differentials, from the 
preceding expression of y, 

(t-4) /d X 

" 



-Xt i , , 

} l~dr; + TT"" I dT^) H 

X is a function of t, and of the constants, c, c , c", &c. and since these 
constants are functions of 6, X is a function of t and of 6, which we can 
represent by <f> (t, 6). The expression of y is by Taylor s Theorem 
the developement of the function <p (t, 6 + t <5), according to the powers 
of t 6. We have therefore y = <p (t, t). Whence we shall have y by 
changing in X, 6 into t. The problem thus reduces itself to determine 
X in a function of t and 6, and consequently to determine c, c , c", &c. 
in functions of 6. 

To solve this problem, let us resume the equation 

y = X + (t 6) . Y + (t /) 2 . Z + &c. 

Since the constant 6 is supposed to disappear from this expression of y, 
we shall have the identical equation 

. . .(a) 



Applying to this equation the reasoning which we employed upon 

= k + k t + k" t 2 + &c. 

we perceive that the coefficients of the successive powers of t 6 ought 
to be each zero. The functions X, Y, Z, &c. do not contain 6, inasmuch 
as it is contained in c, c , &c. so that to form the partial differences 

(i?) , ( 1X> , (*?5 > &c. it is sufficient to make c, c , &c. vary in 
V d 6 ) \ d / Yd* * / - 

these functions, which gives 

X 
d d 



_ (.c ( xc ,} 

- \d c )d 6 H Vd c ) d 4 + \d c") d d 



BOOK I.] NEWTON S PRINCIPIA. 99 

a YX /a YX dc /aY\c /a 



\c , / N 
Vdo + \dc")~dl 

&C. zr &C. 

Again, it may happen that some of the arbitrary constants c, c , c", &c. 
multiply the arc t in the periodic functions X, Y, Z, &c. The differentia 
tion of these functions relatively to 6, or, which is the same thing, relatively 
to these arbitrary constants, will develope this arc, and bring it from without 

the symbols of the periodic functions. The differences ( , --), ( 

\tl D / \ 

V &c. will be then of this form : 



&C. 

X , X", Y , Y", Z , Z", &c. being periodic functions of t, and containing 
moreover the arbitrary constants c, c , c", &c. and their first differences 
divided by d 6, differences which enter into these functions only under a 
linear form ; we shall have therefore 



= Y/ + Y// + ( fc ~ v Y " 

TIT = z + * z" + (t o z- 
ate, 

Substituting these values in the equation (a) we shall have 
= X + 6 X ;/ Y 
+ (t 6) iY + 6 Y" + X" 2 Z} 
+ (t 6) MZ + * Z" + Y" 3 S} + Sec. ; 

whence we derive, in equalling separately to zero, the coefficients of the 
powers of t 6, 

= X + X" Y 

= Y + & Y" + X" 2 Z 

= Z + 0Z" + Y" 3 Sj 

&c. 

.G 2 



100 A COMMENTARY ON [SECT. XL 

If we differentiate the first of these equations, i 1 times successively 
relatively to t, we shall thence derive as many equations between the 
quantities c, c , c", &c. and their first differences divided by d 6. Then 
integrating these new equations relatively to 6, we shall obtain the con 
stants in terms of 6. 

Inspection alone of the first of the above equations will almost always 
suffice to get the differential equations in c, c , c", &c. by comparing se 
parately the coefficients of the sines and cosines which it contains. For 
it is evident that the values of c, c , &c. being independent of t, the dif 
ferential equations which determine them, ought, in like manner, to be in 
dependent of it. The simplicity which this consideration gives to the pro 
cess, is one of its principal advantages. For the most part these equations 
will not be integrable except by successive approximations, which will 
introduce the arc 6 out of the periodic symbols, in the values of c, c , &c. 
at the same time that this arc does not enter the rigorous integrals. But 
we can make it disappear by the following method. 

It may happen that the first of the preceding equations, and its i 1 
differentials in t, do not give a number i of distinct equations between the 
quantities c, c , c", &c. and their differences. In this case we must have 
recourse to the second and following equations. 

When we shall have thus determined c, c , c", &c. in functions of d, 
we shall substitute them in X, and changing afterwards 6 into t, we shall 
obtain the value of y, without arcs of acircle^or free from periodic symbols, 
when that is possible. 

510. Let us now consider any number n of differential equations. 



o = ^-* r + P + Q ; 
&c. 

P, Q, P , Q being functions of y, y , &c. of their differentials to the order 
i 1, and of the sines and cosines of angles increasing proportionally 
with the variable t, whose difference is constant. Suppose the approximate 
integrals of these equations to be 

y - X + t Y + t 2 Z + t 3 S + &c. 

y = X, + t Y, + t 2 Z, + t 3 S, + &c. 

X, Y, Z, &c. X,, Y ; , Z,, &c. being periodic functions of t and containing 
i n arbitrary constants c, c , c", &c. We shall have as in the preceding 
No. 



BOOK I.] NEWTON S PRINCIfiA; 101 

= X + dX" Y; 

= Y + 6 Y" + X" 2 Z; 

= Z + 6 Z" + Y" 3 S ; 

&c. 
The value of y will give, in like manner, equations of this form 

= X/ + *X," Y,; 

= Y/ + 0Y," + X/ g Z /; 

&c. 

The values of y", y ", &c. will furnish similar equations. We shall 
determine by these different equations, selecting the most simple and 
approximable, the values of c, c , c", &c. in functions of 6. Substituting 
these values in X, X , &c. and then changing 6 into t, we shall have the 
values of y, y , &c. independent of arcs free from periodic symbols when 
that is possible. 

511. Let us resume the method already exposed in No. 506. It theucc 
results that, if instead of supposing the parameters c, c , c", &c. constant, 
we make them vary so that we have 

d c = a d t [F Q + F Q + &c} ; 
d c = d t SH Q + H Q + &c.J ; 
we shall always have the i n integrals of the order i 1, 

c = V; c? = V; c" = V" ; &c. 

as in the case of a = 0. Whence it follows that not only the finite in 
tegrals, but also all the equations in which these enter the differences 
inferior to the order i, will preserve the same form, in the case of 
a = 0, and in that where it is any quantity whatever; for these equations 
may result from the comparison alone of the preceding integrals of the 
order i 1. We can, therefore, in the two cases equally differentiate 
i 1 times successively the finite integrals, without causing c, c , &c. to 
vary ; and since we are at liberty to make all vary together, there will 
thence result the equations of condition between the parameters c, c , &c. 
and their differences. 

In the two cases where a = 0, and a = any quantity whatever, the 
values of y, y , &c. and of their differences to the order i 1 inclusively, 
are the same functions of t and of the parameters c, c , &c. Let Y be any 
function of the variables y, y , y", &c. and of their differentials inferior to 
the order i 1, and call T the function of t, which it becomes, when we 
substitute for these variables and their differences their values in t. We 
can differentiate the equation Y = T, regarding the parameters c, c , &c. 
constant ; we can only, however, take the partial difference of Y relatively 

G3 



102 



A COMMENTARY ON [SECT. XL 

to one only or to many of the variables y, y , &c. provided we suppose 
what varies with these," to vary also in T. In all these differentiations, the 
parameters c, c , c", :c. may always be treated as constants ; since by 
substituting for y, y , &c. and their differences, their values in t, we shall 
have equations identically zero in the two cases of" nothing and of a any 
quantity whatever. 

When the differential equations are of the order i 1, it is no longer 
allowed, in differentiating them, to treat the parameters c, c , &c. as con 
stants To differentiate these equations, consider the equation <p = 0, 9 
bein- a differential function of the order i - 1, and which contains the 
parameters c, c , c", &c. Let d f be the difference of this function taken 
in regarding c, c , &c. constant, as also the differences d > 1 y, d - y , &c. 
Let S be the coefficient of & in the entire difference of f. Let S 
be the coefficient of ^ in this same difference, and so on. The e, ua- 
tion 9 = when differentiated will give 



Substituting for & its value - d t IP + . QJ ; for i* value 
_ d t {P + Q S &c. we shall have 



_ d t JS P + S F + &c.} d t [S Q + S Q + &c.} . (t) 
In the supposition of = 0, the parameters c, c , c", &c. are constant. 
We have thus 

= a ? d t S P + S F + &c.} 

If we substitute in this equation for c, c , c", &c. their values V, V, V-, 
&c. we shall have differential equations of the order i - 1 , without arbi- 
traries, which is impossible, at least if this equation is to be id,: 
nothing. The function 

3 p d t {S P + S F + &c.J 

becoming therefore identically nothing by reason of equations c : : V, 
c - V &c. and since these equations hold still, when the parameters 
c, c", C", &c. are variable, it is evident, that in this case, the preceding 



BOOK L] NEWTON S PRINCIPIA. 103 

function is still identically nothing. The equation (t) therefore will be 
come 



a d t {S Q + S Q + &c.} ....... ( X ) 

Thus we perceive that to differentiate the equation <p = 0, it suffices to 
vary the parameters c, c , &c. in <p and the differences d 1 - 1 y, d i ~ 1 y , 
&c. and to substitute after the differentiations, for a Q, a Q , &c. the 

d v d v 
quantities^, -4. , &c. 

Let 4 = 0, be a finite equation between y, y , Sec. and the variable t. If 
we designate by d 4, d z 4, &c. the successive differences of 4, taken in 
regarding c, c , &c. as constant, we shall have, by what precedes, in that 
case where c, c , &c. are variable, these equations : 

4 = 0; 54 = 0; a 2 4 = ...... a 1 - 1 4 = o ; 

changing therefore successively in the equation (x) the function <p into 4, 
d 4> ^ 2 4> &c. we shall have 



=(T \ 

\u c / 



d 



Thus the equations 4 = 0, 4 = 0, &c. being supposed to be the n 
finite integrals of the differential equations 
d ! v 



d t 1 
&c. 

we shall have i n equations, by means of which we shall be able to de 
termine the parameters c, c , c", &c. without which it would be necessary 
for that purpose to form the equations c = V, c = V, &c. But when 
the integrals are under this last form, the determination will be more 
simple. 

512. This method of making the parameters vary, is one of great utility 

G3 



A COMMENTARY ON [SECT. XI. 

in analysis and in its applications. To exhibit a new use of it, let us take 
the differential equation 
d v 

= TP + p 

P being a function of t, y, of their differences to the order i ], and of 
the quantities q, q , &c. which are functions of t. Suppose we have the 
finite integral of this differential equation of the supposition of q, q , &c. 
being constant, and represent by p = 0, this integral, which shall contain 
i arbitraries c, c , &c. Designate by d <p, 3 2 p, 8 3 p, &c. the successive differ 
ences of p taken in regarding q, q , &c. constant, as also the parameters 
c, c , c", &c. If we suppose all these quantities to vary, the differences of 
p will be 



making therefore 

= (d-D d + CH) o " + * + d q + -J ^ + 



a p will be still the first difference of <p in the case of c, c , &c. q, q , &c. 
being variable. If we make, in like manner, 



9 z 9) & 3 P) ..... ^ 5 p will likewise be the second, third, &c. differences of 
<p when c, c , &c. q, q , &c. are supposed variable. 

Again in the case of c, c , &c. q, q , &c. being constant, the differential 
equation 

d v 

= Si? + p 

is the result of the elimination of the parameters c, c , &c. by means of 
the equations p = 0, 8 <p = 0, 8 2 p = 0, . . . . d ! p = 0. Thus, these 
last equations still holding good when q, q , &c. are supposed variable, the 
equation <p = will also satisfy, in this case, the proposed differential 
equation, provided the parameters c, c , &c. are determined by means 
of the i preceding differential equations ; and since their integration 
gives i arbitrary constants, the function <p will contain these arbitraries, 
and the equation <p = will be the complete integral of the proposed 
equation. 



BOOK I.] NEWTON S PRINCIPIA. 105 

This method, the variation of parameters, may be employed with ad 
vantage when the quantities q, q , &c. vary very slowly. Because this 
consideration renders the integration by approximation of the differential 
equations which determine the variables c, c , c", &c. in general much 
easier. 

513. Second Approximation of Celestial Motions. 

Let us apply the preceding method to the perturbations of celestial 
motions, in order thence to obtain the most simple expressions of their 
periodical and secular inequalities. For that purpose let us resume the 
differential equations (1), (2), (3) of No. 471, which determine the relative 
motion of p about M. If we make 

R = l 



+ y 2 + z 2 )^ ( X " 2 + y" 2 + z" 2 )* 



-4- <tc - 
r oii^" 

t* 

X being by the No. cited equal to 
(* 



f (x" _ x ) 2 + (y" yT + (z" z ) 2 } B 
If, moreover, we suppose M + ^ m and 



i" x) 2 +(y" 
r + &c. 



s = V x 2 + y 2 + z 1 

S = V x 2 + y /2 + z 
we shall have 



d 2 z inz 
" dt 2 



(P) 



_ 

The sum of these three equations multiplied respectively by d x, d y, d z 
gives by integration 



2m m 
+ ~ 



~ -- ~ (Q) 



the differential d R being only relative to the coordinates x, y, z of the 
body ft, and a being an arbitrary constant, which, when R = 0, becomes 
by No. 499, the semi-axis major of the ellipse described by ft about 
M. 



106 A COMMENTARY ON [SECT. XI. 

The equations (P) multiplied respectively by x, y, z and added to the 
integral (Q) will give 



We may conceive, however, the perturbing masses /, /// , c. multi 
plied by a coefficient , and then the value of g will be a function of the 
time t and of . If we develope this function according to the powers of a, 
and afterwards make a = 1, it will be ordered according to the powers 
and products of the perturbing masses. Designate by the characteristic 
8 when placed before a quantity, this differential of it taken relatively to , 
and divided by d . When we shall have determined < g in a series or 
dered according to the powers of a, we shall have the radius g by multi 
plying this series by d , then integrating it relatively to , and adding to 
the integral a function of t independent of , a function which is evidently 
the value of g in the case where the perturbing forces are nothing, and 
where the body p describes a conic section. The determination of g re 
duces itself, therefore, to forming and integrating the differential equation 
which determines d g. 

For that purpose, resume the differential equation (R) and make for the 
greater simplicity 

d Rx /d R 



differentiating this relatively to , we shall have 



Call d v the indefinitely small arc intercepted between the two radius- 
vectors g and g + d g ; the element of the curve described by //. around M 
will be V dg 2 + g*d\\ We shall thus have 

clx 2 + dy 2 + dz 2 d z + g 2 d v 2 , 
and the equation (Q) will become 



dt 2 g a 

Eliminating from this equation by means of equation (R) we shall 

ft 

have 

^Tt 1 " : Tt^" + T + s R 

whence we derive, by differentiating relatively to a, 

d t 2 d t 2 r* *~ s a ~~ s 



BOOK I.] NEWTON S PRINCIPIA 107 

If we substitute in this equation for ^-^ its value derived from equa 
tion (S), we shall have 



By means of the equations (S), (T), we can get as exactly as we wish the 
values of B g and of d v. But we must observe that d v being the angle 
intercepted between the radii g and g + d & the integral v of these angles 
is not wholly in one plane. To obtain the value of the angle described 
round M, by the projection of the radius-vector g upon a fixed plane, de 
note by v, , this last angle, and name s the tangent of the latitude of ^ above 

this plane ; then g (I + s 2 ) ~ will be the expression of the projected ra 
dius-vector, and the square of the element of the curve described by p, 
will be 

r+V 2 + df2 + (iT^r 5 

But the square of this element is also g 2 d v 2 + d g 2 ; therefore we have, 
by equating these two expressions 



., 



We shall thus determine d v y by means of d v, when s is known. 

If we take for the fixed plane, that of the orbit of p at a given epoch, 

s an{ i J w i]l evidently be of the order of perturbing forces. Neglecting 
d v 

therefore the squares and the products of these forces, we shall have 
v = v, . In the Theory of the planets and of the comets, we may neglect 
these squares and products with the exception of some terms of that 
order, which particular circumstances render of sensible magnitude, and 
which it will be easy to determine by means of the equations (S) and (T). 
These last equations take a very simple form, when we take into account 
the first power only of the disturbing forces. In fact, we may then con 
sider 8 i and d v as the parts of g and v due to these forces ; d II, d. g R 
are what R and g R become, when we substitute for the coordinates of 
the bodies their values relative to the elliptic motion : We may designate 
them by these last quantities when subjected to that condition. The 
equation (S) thus becomes, 

= + =-- + 2/rf R + , R . 



108 A COMMENTARY ON [SECT. XL 

The fixed plane of x, y being supposed that of the orbit of ^ at a given 
epoch, z will be of the order of perturbing forces : and since we may 
neglect the square of these forces, we can also neglect the quantity 

Z \dz) Moreover, the radius g differs only from its projection by quan 

tities of the order z 2 . The angle which this radius makes with the axis 
of x, differs only from its projection by quantities of the same order. 
This angle may therefore be supposed equal to v and to quantities nearly 
of the same order 

x = cos. v ; y = g sin. v ; 
whence we get 

d R 



and consequently g . R = s ^fr li is eas y to perceive by differentia 

tion, that if we neglect the square of the perturbing force, the preceding 
differential equation will become, by means of the two first equations (P) 

/y^ / R + Kffil-y/ d fr/ H +,(ffi } 

/x d y y d xx 

v ai ) 

In the second member of this equation the coordinates may belono- to 
elliptic motion ; this gives ^7? ( constant and equal to V~m a(l e 2 ), 

a e being the excentricity of the orbit of p. If we substitute in the ex 
pression of 8 for x and y, their values g cos. v and sin. v, and for 

x d y _ v d x _ 

c j t - , the quantity v" ,- a (1 e 2 ) ; finally, if we observe that 

by No. (480) 



m = n 2 a 
we shall have 



(" a cos. v/n d t . s sin. v { 2fd R + s (-.} \ 

Vd e - 



} ) 

e - V 

j 



(X) 
-sin.v/ndt. e cos.v2/rfR + f () 

o o ^: -- - - =^ - 5 - 

m V 1 e 2 

The equation (T) gives by integration and neglecting the square of 
perturbing forces, 



2 g d . a g + d s . d s 3 a rr , , 2 a , /d Rx 

- - i PT-= - + - - // n d t . d R H -- fn d t. g ( , ) 

a * n d t ^ m J - m J s \ d / 



.,,. 



BOOK I.] NEWTON S PRINCIPIA. 109 

This expression, when the perturbations of the radius-vector are known, 
will easily give those of the motion of p in longitude. 

It remains for us to determine the perturbations of the motion in lati 
tude. For that purpose let us resume the third of the equations (P): 
integrating this in the same manner as we have integrated the equation 
(S), and making z = f 8 s, we shall have 

r -, . . /d R\ . r j ^ Vd R\ 

a cos. vyn d t.^sin. v [. ) asm. vyndt.gcos. vf-r ) 

a s = - dz/ - Uz ; (Z) 

m v 1 e 2 

6 s is the latitude of /a above the plane of its primitive orbit: if we wish 
to refer the motion of /A to a plane somewhat inclined to this orbit, by 
calling s its latitude, when it is supposed not to quit the plane of the 
orbit, s + 5 s will be very nearly the latitude of & above the proposed 
plane. 

514. The formulas (X), (Y), (Z) have the advantage of presenting the 
perturbations under a finite form. This is very useful in the Cometary 
Theory, in which these perturbations can only be determined by quad 
ratures. But the excentricity and inclination of the respective orbits of 
the planets being small, permits a developement of their perturbations 
into converging series of the sines and cosines of angles increasing pro 
portionally to the time, and thence to make tables of them to serve for 
any times whatever. Then, instead of the preceding expressions of 8 g, 
8 s, it is more commodious to make use of differential equations which 
determine these variables. Ordering these equations according to the 
powers and products of the excentricities and inclinations of the orbits, 
we may always reduce the determination of the values of B g, and of 8 s 
to the integration of equations of the form 



equations whose integrals we have already given in No. 509. But we 
can immediately reduce the preceding differential equations to this simple 
form, by the following method. 

Let us resume the equation (R) of the preceding No., and abridge it 
by making 



It thus becomes 





110 A COMMENTARY ON [SECT. XI. 

In the case of elliptic motion, where Q = 0, g 2 is by No. (488) a func 
tion of e cos. (n t + t *), a e being the excentricity of the orbit, and 
n t + e -a the mean anomaly of the planet p. Let e cos. (n t + w ) 
= u, and suppose 2 = <p (u) ; we shall have 



In the case of disturbed motion, we can still suppose p, 2 = <f> (u), but 
u will no longer be equal to e cos. (n t + t *r). It will be given by 
the preceding differential equation augmented by a term depending upon 
the perturbing forces. To determine this term, we shall observe that if 
we make u = 4/ (g 2 ) we shall have 



4/ (e 2 ) being the differential of -fy (e. 2 ) divided by d.* 2 and ^" (g 2 ) the 

d 2 ?* 
differential of -4/ (f 2 ) divided by d.f 2 . The equation (R ) gives -jf^- 

equal to a function of g plus a function depending upon the perturbing 
force. If we multiply this equation by 2 f d f, and then integrate it, we 

2 1 2 

shall have ^U f- equal to a function of g plus a function depending upon 

d 2 . e 2 e 2 d e 2 . 

the perturbing force. Substituting these values of ~ 2 and of , 8 - in 

the preceding expression of -. - + n 2 u, the function of & which is in 

dependent of the perturbing force will disappear of itself, because it is 
identically nothing when that force is nothing. We shall therefore have 

d 2 u d 2 . e 2 p 2 de 2 

the value of - - + n 2 u by substituting for , and , 2 , the parts 

Ci 1 Q C 

of their expressions which depend upon the perturbing force. But re 
garding these parts only, the equation (R ) and its integral give 

d2 - ?2 - 20- 

"" 



Wherefore 

. d s- 



Again, from the equation u = <p ( 2 ), we derive d u 2 g d g ty (f 2 ) ; 
this f * = <p (u) gives 2 p, d f = d u. f f (u) and consequently 



BOOK 1.1 NEWTON S PRINCIPIA. Ill 

4 (f 2 ) = p~fifi* 

Differentiating this last equation and substituting <ff (u) for - j ^ , we 
shall have 



p" (u) being equal to * ^ , in the same way as <f> (u) is equal to 

. P u/) . . This being done ; if we make 
d u 

u = e cos. (n t + <ar ) 4* ^ u 
the differential equation in M will become 






and if we neglect the square of the perturbing force, u may be supposed 
equal to e cos. (n t + 2 ), in the terms depending upon Q. 

The value of - found in No. (485) gives, including quantities of the 
a 

order e 3 

, = .{l + e -n(l-f e)-n -f u } 

whence we derive 

^ = a a |l + 2e 2u(l i e 2 ) u 2 u 3 j = p (u). 
If we substitute this value of p (u) in the differential equation in d u, 
and restore to Q its value 2 / d R + g (-jr) and e cos - ( n l + s w ) 
for u, we shall have including quantities of the order e 3 , 



i-fl + 4 e 2 ecos. (nt + ) --e cos. (2 n t + 2 a 

a z 4 4t 



When we shall have determined 5 u by means of this differential equa- 



112 A COMMENTARY ON [SECT. XI. 

tion, we shall have 3 g by differentiating the expression of g, relative to 
the characteristic <3, which gives 

f 3 9 1 

dg = adu< 1 + -e 2 +2ecos. (n t + )+ - e 2 cos.(2nt+ 2s 2tr) V. 

This value of 5 g will give that of d v by means of formula (Y) of the 
preceding number. 

It remains for us to determine d s ; but if we compare the formulas (X) 
and (Z) of the preceding No. we perceive that d g changes itself into 8 s 

by substituting (^-) for 2fdR + g frp-1 in its expression. Whence 

it follows that to get d s, it suffices to make this change in the differential 
equation in (5 u, and then to substitute the value of 5 u given by this equa 
tion, and which we shall designate by d u , in the expression of 8 g. Thus 
we get 

o =rjr + **** 

"a 2 ! 1 + 4 et ~~ e cos. (n t+ e w) ;j- e*cos.(2nt-f 2 2 



3s= aSu 1 1 + ~e* +2 e cos. (nt + s w)+ ^e 2 cos.(2nt-f 2e 2) j 

The system of equations (X 7 ), (Y), (Z ) will give, in a very simple 
manner, the perturbed motion of IL in taking into account only the first 
power of the perturbing force. The consideration of terms due to this 
power being in the Theory of Planets very nearly sufficient to determine 
their motions, we proceed to derive from them formulas for that purpose. 

515. It is first necessary to develope the function R into a series. If 
we disregard all other actions than that of ,- upon ^ , we shall have by (513} 
R _. .^(xx +yy +zzO __ ^ ___ 

(x /2 + y /2 + z 2 )^ f(x x) 2 + (y _ y) 2 + (z z) 2 ^ 

This function is wholly independent of the position of the plane of x, 
y ; for the radical V (x x) 2 + (y y) 2 + (z z) 2 , expressing the 
distance of n, ,/, is independent of the position ; the function x 2 + y 2 
-f z 2 + x z + y 2 + T! 2 2 x x 2 y y 2 z z is in like manner in 
dependent of it. But the squares x 2 + y 2 + z 2 and x /2 + y /2 + z 2 
of the radius- vectors, do not depend upon the position ; and therefore the 
quantity x x + y y + z z does not depend upon it, and consequently 



BOOK I.] NEWTON S PRINCIPIA. 113 

R is independent of the position of the plane of x, y. Suppose in this 
function 



we shall then have 
K - ^gg /cos - 

At ~ 



x = f cos. v ; y = f sin. v ; 
x = g cos. v ; y P sin. \ f ; 



^ 

(/ 2 + z 2 ) 2 g z2 f cos. ( v v) + g 2 + (z z) 2 ] * 

The orbits of the planets being almost circular and but little inclined 
to one another, we may select the plane of x, y, so that z and z may be 
very small. In this case g and g f are very little different from the semi- 
axis-majors a, a of the elliptic orbits, we will therefore suppose 

g = a(l + u,); f = a (l + u/); 

u, and u/ being small quantities. The angles v, v differing but little 
from the mean longitudes n t + , n t + t , we shall suppose 

v = n t + s + v,; v = n t + + v/; 

v and v/ being inconsiderable. Thus, reducing R into a series ordered 
according to the powers and products of u,, v,, z, u/, v/, and z , this series 
will be very convergent. Let 






~ 9 cos. (n t n t + J {a 2 2 a a cos. (n t n t + i e)-f a 2 } ~ 

= g A w + A < cos. (n 7 1 - n t + - -r- A cos. 2 (n t n t +/ *) 

+ A W cos. 3 (n 7 t n t + e ) + &c. ; 

We may give to this series the form 2 A W cos. i (n t n t + i _ *), 
the characteristic 2 of finite integrals, being relative to the number i, and 
extending itself to all whole numbers from i = co to i = oo ; the value 
i = 0, being comprised in this infinite number of values. But then we 
must observe that A <- = A (i) . This form has the advantage of serving 
to express after a very simple manner, not only the preceding series, but 
also the product of this series, by the sine or the cosine of any angle 
ft + &; for it is perceptible that this product is equal to 



This property will furnish us with very commodious expressions fcr 
the perturbations of the planets. Let in like manner 

[a * 2 a a cos. (n t n t -f- 1 ) -f a 2 ] ~" * 

= ^ 2 B cos. i (n t n t + t) ; 

B<- > being equal to B <". This being done, we shall have by (483) 
VOL. II. 



114 A COMMENTARY ON [SECT. XI. 

ttf 
R = - . 2 A W cos. i (n t n t + e) 



+ u, 2 a( d d A a -)cos. i (n t _ n t + - ) 
s - i (n t n t + i 



u 

-- (v/ v,) 2 . i A sin. i (n t n t + % -- 



- . u,. 2 -a 2 -s- i (n t - n t + - 

S .i(n t nt + , ,) 

s. i n t - n t + * - ,) 



( v / - v ) u / 2 J a/ - Sin< i (n t - n t + . - 



~ (v/ v,) 2 . 2 . i * A (l > cos. i (n t n t + s 

i 1 

/// z z 3 /i a T! z 
+ , s --- f^-4- cos. (n t n t + ) 

/ fy _ y\Z 

+ ^ 2 B W cos. i (n t n t + s 

+ &c. 

If we substitute in this expression of R, instead of u /5 u/, v /} v/, z and z 7 , 
their values relative to elliptic motion, values which are functions of sines 
and cosines of the angles n t + s, n t + t r and of their multiples, R will 
be expressed by an infinite series of cosines of the form < k cos. (i n t 
i n t + A), i and i being whole numbers. 

It is evident that the action of (J>", (* ", &c. upon p will produce in R 
terms analogous to those which result from the action of //, and we shall 
obtain them by changing in the preceding expression of R, all that relates 
to /* , in the same quantities relative to &"> i"/", &c. 

Let us" consider any term (i! k cos. (i n t i n t + A) of the expres 
sion of R. If the orbits were circular, and in one plane we should 
have i = i. Therefore i cannot surpass i or be exceeded by it, except 
by means of the sines or cosines of the expression for u /} v /9 z, u/, v/, z 
which combined with the sines and cosines of the angle n t nt + l/ f 



BOOK I.] NEWTON S PRINCIPIA. 115 

and of its multiples, produce the sines and cosines of angles in which i 
is different from i. 

If we regard the excentricities and inclinations of the orbits as veiy 
small quantities of the first order, it will result from the theorems of 
(48 1 ) that in the expressions of u /5 v,, z or g s, s being the tangent of the 
latitude of p, the coefficient of the sine or of the cosine of an angle such 
as f. (n t + 2), is expressed by a series whose first term is of the order f ; 
second term of the order f + 2 ; third term of the order f + 4 and so 
on. The same takes place with regard to the coefficient of the sine or of 
the cosine of the angle f (n t + /) in the expressions of u/, v/, z . Hence 
it follows that i, and i being supposed positive and i greater than i, the 
coefficient k in the term m k cos. (i n t i n t + A) is of the order 
i i, and that in the series which expresses it, the first term is of the 
order V i the second of the order V i -f 2 and so on ; so that the 
series is very convergent. If i be greater than i , the terms of the series 
will be successively of the orders i i , i V -j- 2, &c. 

Call -a the longitude of the perihelion of the orbit of p and 6 that of its 
node, in like manner call ** the longitude of the perihelion of ,/, and 
that of its node, these longitudes being reckoned upon a plane inclined 
to that of the orbits. It results from the Theorems of (481), that in the 
expressions of u /5 v,, and z, the angle n t + s is always accompanied by 
or by 6 1 and that in the expressions of u/, v/, and z , the angle 
n t + t is always accompanied by , or by 6 ; whence it follows 
that the term (t! k cos. (i n t i n t + A) is of the form 

y! k cos. (i n 7 1 i n t -f- i e is g - g 7 J g" 6 g" </), 
g, g , g", g" being whole positive or negative numbers, and such that 
we have 

= i - i g g g" g" . 

It results also from this that the value of R, and its different terms are 
independent of the position of the straight line from which the longitudes 
are measured. Moreover in the Theorems of (No. 481) the coefficient of 
the sine and cosine of the angle , has always for a factor the excentricity e 
of the orbit of p ; the coefficient of the sine and of the cosine of the angle 
2 9) has for a factor the square e 2 of this excentricity, and so on. In like 
manner, the coefficient of the sine and cosine of the angle d, has for its 
factor tan. \ tp, <p being the inclination of the orbit of /A upon the fixed 
plane. The coefficient of the sine, and of the cosine of the angle 2 6, has for 
its factor tan. 2 \ <p, and so on. Whence it results that the coefficient k has for 

its factor, e *. e s . tan. g " ( <p ) tan. g/// ( <f> } ; the numbers g, g 7 , g", g" being 

H2 



116 A COMMENTARY ON [SECT. XI. 

taken positively in the exponents of this factor. If all these numbers are 
positive, this factor will be of the order i 7 i, by virtue of the equation 

= i __i_g_g _g"_ g" ; 

but if one of them such as g, is negative and equal to g, this factor 
will be of the order i i + 2 g. Preserving, therefore, amongst the 
terms of R, only those which depending upon the angle i n 7 t i n t are of 
the order i i, and rejecting all those which depending upon the same 
angle, are of the order i i + 2, i i + 4, &c. ; the expression of 
R will be composed of terms of the form 

H e 8. e 7 tan. *" ( ~ p) tan. * ". ( i ?/) cos. (i n t i n t + i * 

_ i f _ g. g . . g". 8 - g 777 . ), 

H being a coefficient independent of the excentricities, and inclinations 
of the orbits, and the numbers g, g , g", g " being all positive, and such 
that their sum is equal to i i. 

If we substitute in R, a (1 + u y ), instead of s , we shall have 
d Rx /d R 



If in this same function, we substitute instead of u 7 , v 7 and z, their values 
given by the theorems of (481), we shall have 

/d RN _ /d R- N . 

Vd v/~J>d J 

provided that we suppose s , and s ^ constant in the differential of 
R, taken relatively to z ; for then u /} v / and z are constant in this differ 
ential, and since we have v = n t -f s + v /} it is evident that the preced 
ing equation still holds. We shall, therefore, easily obtain the values 

and of f^r^V which enter into the differential equations of 



the preceding numbers, when we shall have the value of R developed 
into a series of angles increasing proportionally to the time t. The dif 
ferential d R it will be in like manner easy to determine, observing to vary 
in R the angle n t, and to suppose n 7 t constant ; for d R is the difference 
of R, taken in supposing constant, the coordinates of //, which are func 

tions of n 7 t. 

516. The difficulty of the developement of R into a series, may be 
reduced to that of forming the quantities #, B *\ and their differences 
taken relatively to a and to a 7 . For that purpose consider generally the 
function 

(a 2 2 a a cos. 6 + a 72 ) 



BOOK I.] NEWTON S PRINCIPIA. 117 

and develope it according to the cosine of the angle Q and its multiples. 

If we make ; = a, it will become 




a 



a * i 

a . { ] 2 a cos. A -j- a *} 
Let 

( 1 2 a cos. 4 + a 2 ) ~ 3 = b ^ + b C1 > cos. tf + b to cos. 2 * 

S 6 S 

+ b cos. 3 + &c. 

S 

b (0) , b (l) , b (fi ), &c. being functions of a and of s. If we take the logarith- 

f S S 

mic differences of the two members of this equation, relative to the vari 
able d } we shall have 



1 2 a cos. 6 + a. z % b W + b (1) cosJ+b W cos. 

S S g 

Multiplying this equation crosswise, and comparing similar cosines, we 
find generally 

(i 1) (1 + ^b^-D (i + s 2)ab< 1 - 2 > 

b G) = - s -^ - - - -^ - 



. a 



We shall thus have b (2 >, b \ &c. when b W and b P) are known. 

S B 

If we change s into s + 1, in the preceding expression of (1 2 a cos. & 

~ s 

a 2 ) , we shall have 



(1 2cos. d+u z } "zr^bW + bWcos. 0+b cos.2 

8 + 1 S + l 8 + 1 B + l 

Multiplying the two members of this equation, by 1 2 cos. rf + % 

and substituting for ( 1 2 a cos. + a ! ) ~ its value in series, we shall 
have 

b (c > + b (1 > cos. d + b & cos. 2 + &c. 

S3 S 

= (1 2acos.0+a 2 ) b + b (1 >cos.0 + b^cos. 20 + &C.J 

S + l S+ 1 8 + 1 

whence by comparing homogeneous terms, we derive 

b > = (1 + 2 )b (0_bl i - 1 ) ab^ i + 1 ). 

,,,, r . S+l 8 + 1 S + l 

1 he formula (a) gives 

i(l + 2 )bW (i + sjob 1 -" 



s+l _ S . a 

Tiie preceding expression of b will thus become 



L+J 



S 
H3 



118 A COMMENTARY ON [SECT. XL 

Changing i into i + 1 in this equation we shall have 



_ ___ 

i s + 1 

and if we substitute for b (i + J) its preceding value, we shall have 

s + l 



b + D = _ 1-1 _ _ i+J 
s s ) (l S + l)a 

These two expressions of b (i) and b (i + J) give 

s s 

l. ( i + g .)bc)-2. i - s+ l bo.* ) 

S 

! _ - _ ! _ f M 
s + l (1 T 

substituting for b (i + J) its value derived from equation (u), we shall have 



~ 8 (i^W~ ~ ; (c) 

an expression which may be derived from the preceding by changing i 
into i, and observing that b (i) = b (-i) . We shall therefore have by 
means of this formula, the values of b (0) , b (1) , b (2) , &c. when those of 

s+l s+l s+l 

b ( % b (l \ b (2) , &c. are known. 

a as 

Let X, for brevity, denote the function 1 2 o cos. 6 + a 2 . If we 
differentiate relatively to a, the equation 

X - = b (> + b (1) cos. 6 + b cos. 2 6 + &c. 

88 8 

we shall have 

d b ( ) d b > d b ( 2 ) 

2 s (a cos. 6} X ~ - 1 = A . - 1 f cos. 6 + -j 8 cos. 2 6 + &c. 

2 da da da 

But we have 



a + cos. ^ = 



2 a 
We shall, therefore, have 



7\. _ Q j ~1 *V\JJ* " -| JV*- 

a a ^ d a a a 

whence generally we get 

_ S (1 tt J ^ (i) _ _8 ^ 

da a g + i a 

Substituting for b (i > its value given by the formula (b), we shall have 

6 + 1 

^__i + (i + 2s). 0) 2(i-s+l) 
"dV "" a(] a 2 ) b . W 1 a 2 . + 



BOOK I.] NEWTON S PRINCIPIA. 119 

If we differentiate this equation, we shall have 

2 (i+s) 




1 a 

Again differentiating, we shall get 



d " 



, 





11 

a 2 / da 



<x(l a 2 ) da 2 (11 a 2 ) 8 

d 2 b (i 
4 (i + s) a (3 + a 2 ) 2_il , ni 2(i s+1) 



h 23 ^ 



_ 

(1 a 2 ) 3 a 3 / 1 a 2 da 2 

, 



(1 a 2 ) 2 da (I a 2 ) 3 

Thus we perceive that in order to determine the values of b and 01 

8 

its successive differences, it is sufficient to know those of b w and of b (1) . 

8 8 

We shall determine these two as follows : 

If we call c the hyperbolic base, we can put the expression of X s un 
der this form 

X- 8 = (1 a c flv T ) - 9 (1 C 0V !)-. 

Developing the second member of this equation relatively to the powers of 
c 6 V 1 , and c ~ 6 ^~ l , it is evident the two exponentials c i e V 1 , c i 6 V 1 
will have the same coefficient which we denote by k. The sum of the 
two terms k . c l e v 1 and kc i \/ Ms 2k cos. i 6. This will be the 
value of b (i ) cos. i 0. We have, therefore, b (i) = 2 k. Again the ex- 

8 S 

pression of X - s is equal to the product of the two series 



sa c -i + 1 c8V-i + &c. 

! SB 



multiplying therefore these two together, we shall have when i = 
k = l S 2 a^ 



and in the case of i = 1, 



wherefore 



H I 



120 A COMMENTARY ON [SECT. XL 



b < = 2 



+JO 4 



. . . . 

That these series may be convergent, we must have a less than unity, 
which can always be made so, unless a = a ; a being = - , we have only 

to take the greater for the denominator. 

In the theory of the motion of the bodies -, /. , /t", &c. we have occasion 
to Ivnovv the values of b (0) and of b ^ when s = % and s = f . In these 

8 S 

two cases, these values have but little convergency unless is a small 
fraction. 

The series converge with greater rapidity when s = , and we have 



id) f Ll 2 1 1.1.3 4 1.3 1.1.3.5 6 1.3.5 1.1.3 ..7 

V K V~2.4 a ~4 2A6 a ~^6-2A6^ a ~4^8 273T^10 

" 2 

In the Theory of the planets and satellites, it will be sufficient to take 
the sum of eleven or a dozen first terms, in neglecting the following 
terms or more exactly in summing them as a geometric progression whose 
common ratio is 1 2 . When we shall have thus determined b (0) and 



b n \ we shall have b (0) in making i = 0, and s = in the formula (b), 
and we shall find 



i " 
If in the formula (c) we suppose i = I and s = we shall have 



, - 

By means of these values of b (0) and of b (1) we shall have by the pre- 

i I 

ceding forms the values of b (i) and of its partial differences whatever may 

sT 

be the number i ; and thence we derive the values of b (l) and of its dif 



ferences. The values of b ( ) and of b (1) may be determined very simply, 



BOOK L] NEWTON S PRINCIPIA. 121 

by the following formula? 

b w b () 

b w = ~ . 



Again to get the quantities A <% A l \ &c. and their differences, we 
must observe that by the preceding No., the series 

A ) -f- A ) cos. + A U cos. 2 + &c. 
results from the developement of the function 

a cos. 6 _ i 

-- (a 2 2 a a cos. d + a 2 ) *, 

into a series of cosines of the angle 6 and of its multiples. Making ~ = , 
this same function becomes 



S 

which gives generally 



ACi> = _.b< ; 

a i 

when i is zero, or greater than 1, abstraction being made of the sign. 
In the case of i = ], we have 



We have next 



/dAx \ ,da. 

V da )~ " a da \daJ ; 






But we have -, = ; therefore 
da a 



d b w 

(i S - _ J_ _i_ 

a/~ a /z *da 



d 

and in the case of i = ] , we have 

d b ) 

/dAWx JLJ i ) 

V da ;~ a /2 I da J 

Finally, we have, in the same case of i = I 

d ~ b ti} 
d 2 



/d 2 AWx J_ i 

V d a 2 y ~ a 3 d z ; 



122 A COMMENTARY ON [SECT. XI. 

d 3 b 

/d 3 A (i \ J __ 

V da 3 / a 4 da 3 

&c. 

To get the differences of A (i) relative to a , we shall observe that A w 
being a homogeneous function in a and a , of the dimension 1, we 
have by the nature of such functions, 

/dAx, ,/dAx Am 

a (5 - ) + a ( 17-1 = A (1) ; 
v d a / vda / 

whence we get 



--__ 

da - da 

a , d 



A W\ /d A 

= 2 A " + 4 



, 3 /d 3 A x . fo /d A < 1 K ,d 2 A 0). /d^A^x . 

a ( -j /T~ ) = 6 A I 8 a I j ) 9a ( \ F ) a ( ~i r ) 
\da /3 / \da/ \da 2 / xda j / 

&c. 

We shall get B (i) and its differences, by observing that by the No. pre 
ceding, the series 

i B (0 > + B W cos. 6 + B ^ cos. 2 Q + &c. 
is the developement of the function 

a - 3 (1 2 a cos. 6 + a 2 )"^ 

according to the cosine of the angle 6 and its multiples. But this function 
thus developed is equal to 

a - 3 fb<> + b> cos. d + b cos. 2 6 + &c.) 

II 1 I 

therefore we have generally 



a 
Whence we derive 



db m d b 

; * 



_B^x J_ _|_ ; /d BWx ^ | 
da / a /4< da V d a 2 / ~ a /s d a 2 
Moreover, B (i > being a homogeneous function of a and of a , ot the 
dimension 3 we have 



a 



d a / V d a 



BOOK I.] NEWTON S PRINCIPIA. 123 

whence it is easy to get the partial differences of B w taken relatively to 
a by means of those in a. 

In the theory of the Perturbations of /* , by the action of p, the values 
of A Ci) and of B (1) , are the same as above with the exception of A (il which 

in this theory becomes , -- b (1) . Thus the estimate of the values of 
2i a 

2 

A (i) , B ( , and their differences will serve also for the theories of the two 
bodies /. and fjf. 

517. After this digression upon the developement of R into series, let 
us resume the differential equations (X ), (Y), (Z ) of Nos. 513, 514; and 
find by means of them, the values of 3 g, 8 v, and d s true to quantities 
of the order of the excentricities and inclinations of orbits. 

If in the elliptic orbits, we suppose 

f = a(l + u,); e =a (l+u/): 
v = n t + s + v 7 ; v = n t s + v/; 
we shall have by No. (488) 

u, = e cos. (n t + s -a}; u/ = e cos. (n t + s /) ; 
v, = 2 e sin. (n t + r) ; v/ = 2 e sin. (n t + e */) ; 
n t + , n t + e being the mean longitudes of /*, fi! ; a, a being the semi- 
axis-majors of their orbits ; e, e the ratios of the excentricity to the semi- 
axis-major ; , and lastly r, & being the longitudes of their perihelions. All 
these longitudes may be referred indifferently to the planes of the orbits, 
or to a plane which is but very little inclined to the orbits ; since we ne 
glect quantities of the order of the squares and products of the excen 
tricities and inclinations. Substituting the preceding values in the ex 
pression of R in No. 515, we shall have 

R = ~ 2 A cos. i ( n t n t + i 7 f) 



e cos.Ji (n t n t + f t) + n t + t 



e cos.{i (n 7 t n t + e e) + n t + t */}; 

the symbol 2 of finite integrals, extending to all the whole positive and 
negative values of i, not omitting the value i = 0. 
Hence we obtain 



124 A COMMENTARY ON [SECT. XI. 




d 

+ n t + * }; 

the integral sign 2 extending, as in what follows, to all integer positive 
and negative values of i, the value i = being alone excepted, because 
we have brought from without this symbol, the terms in which i = : /* g 
is a constant added to the integral/" d R. Making therefore 



. . 

i(n n ) n 



. 
r i (n n ) n I V d 

taking then for unity the sum of the masses M + /<, and observing that 
(237) M + ^ = n 2 , the equation (X ) will become 



BOOK I.] NEWTON S PRINCIPIA. 125 

+ n 2 (. C e cos. (n t -f i -a] 
+ ri 2 v/ D e cos. (n t + ) 

+ n - fi! 2 C W e cos. i (n t n t + sf i) + n t + wj 
+ n V 2 D (i) e cos.i (n t n t + e s) + n t + }; 
and integrating 

to 



uf I \ d a / IM j / / . / \ 

~ -- n 2 2 . ^- 7-5- 5 cos. i (n t n t + ) 

2 i 2 (n n ) 2 n 2 

+ . f e cos. (n t + -a] + & f / e sin. (n t + ) 
/ \ / / \ 

/ / 
C . n t . e sin. (n t + w ) D . n t. e sin. (n t + i a/) 



+ ^ 2 li(n--n}*-n* 6/ sji (n/ 1 " " n t +i/ ~ ) +n t + g ~ w/ ? 
f x and f/ being two arbitraries. The expression of d in terms 6 u, found 
in No. 514 will give 



_ 

i^-l 1 T^^ 

//re cos. (n t + ^) // f e cos. (n t + 2 ^ ) 

+ ,v/ C n t e sin. (n t+ e ) + ^ D n t e sin. (n t + */) 



r ,,, 

, 2 )J \da/^n n x ^ _ Cj 1 i 

^ I i 2 (n ri ) 2 n 2 Ji (n n ) n} 2 n s ) 

X e cos. i (n x t n t + e s) + n t + } 

^ - n 2 2 . U(n _ n ^ nr _ n2 e / cos. {i(n t-n t+ e _)+n t+-r ], 

f and f being arbitrary constants independent of f /5 f/. 

This value of 5 g, substituted in the formula (Y) of No. 513 will give 3 v 
or the perturbations of the planet in longitude. But we must observe that 
n t expressing the mean motion of /*, the term proportional to the time, 
ought to disappear from the expression of 8 v. This condition determines 
the constant (g) and we find 



126 A COMMENTARY ON [SECT. XI. 

We might have dispensed with introducing into the value of d g the 
arbitraries f, f/, for they may be considered as comprised in the elements 
e and -a of elliptic motion. But then the expression of 8 v would include 
terms depending upon the mean anomaly, and which would not have 
been comprised in those which the elliptic motion gives : that is, it is more 
commodious to make these terms in the expression of the longitude dis 
appear in order to introduce them into the expression of the radius-vector > 
we shall thus determine f, and f/ so as to fulfil this condition. Then if we 

/d A (i - 1} \ . /d A ( i- J) .\ 

substitute for a ( 5 ; ) its value A (l-1) a ( -j- ) , we shall 
v d a / \ d a / 

have 

/ 

r n 2 I 
V 



A da , 

d A >\ 


i iu> vuiuc r\ 

/d 2 A (0) \ 

In 3 ! .V 


da/ 


* V da 2 / 



Moreover let 
, M , 3 n . m , i 2 (n n 7 ) . {n + i (n n )] 3 n 2 



. d 2 

x a 



n n i2( n _n ) 2 n 2 

d A x 2 n , ,) 2 nE 



f 
l 



. (i) 
a 2 



n 2 {n i(n n )} 2 



(i _ ]) (2 i_ 1) n a A*- + (i - 1) n a 
G w 



2 [n i (n n )J 

2 n 2 D (i) 
n 2 Jn i (n n )} 2 



BOOK I.] NEWTON S PRINCIPIA. 127 

and we shall have 



, d A x 2n 

a -~ aA 



2 i*( n n ) f n 2 

cos. i (n t n t + t r s) 

ft fe cos. (n t + + ) /et f e cos. (n t + i ) 
+ p C.ntesin. (nt + g w) + /Dn te sin. (n t + i ) 






C o a/ 2 /dA (l) \ 2n ) -\ 

. i . 2n s ia s (-j \-\ iaAj f . . 

J n 2 . m . I \ d a / n n 7 I > sin. i 

av=2) rr aAW + - J-J , TT-T i 

2 (_i(n n )* i (n n ) . U . (n n x ) 2 n 2 ] ) 

(n t n t + e i) 
+ (t! . C . n t . e cos. (n t + s *?} -f- p D . n t . e cos. (n t + e */) 

f F ^ 

I , esin. Ji(n t n t + t t) + nt + i*}~] 

n i (n n ) 

i , ../ \? J ^ * 



n i (n nO~ ~ 

tlie integral sign 2 extending in these expressions to all the whole positive 
and negative values of i, with the value i = alone excepted. 

Here we may observe, that even in the case where the series represent 
ed by 

2. A (i) cos. 5 (n t n t + i e) 

^\ 

is but little convergent, these expressions of and of d v, become con- 

a 

vergent by the divisors which they acquire. This remark is the more 
important, because, did this not take place, it would have been impossible 
to express analytically the mutual perturbations of the planets, of whic 
the ratios of their distances from the sun are nearly unity. 

These expressions may take the following form, which will be useful to 
us hereafter. Let 

h r= e sin. -a ; h = e sin. / ; 
1 = e cos. w, 1 = e cos. -a ; 
then we shall have 



h 



^ (hf + h f) cos. (n t + s) v! (1 f + \ f) sin. (n t + t) 



128 A COMMENTARY ON [SECT. XL 

+ ^ {1 C + l D] n t sin. (n t + e) [h C + h D}n t cos. (n t + 



2 




= ,. _L_. A + an . . n , 

2 li(n n ) 2 1 (n n ) {i 2 . (n _ n ) *~ n*} j 

sin. i (n t n t + t s) 
[h C + h D}. n t . sin. (n t + e)+/* {1 . C + l . D} n t. cos. (n t + s) 



+ nX.J"-^f in U(n/t ~ nt + / ~ e) + nt+<} Is 

^_hF^+h^ cos ^ n/t _ n ;/ _ r 

V. n i(n iv ) 

Connecting these expressions of d and 3 v with the values of and v 
relative to elliptic motion, we shall have the entire values of the radius- 
vector of /*, and of its motion in longitude. 

518. Now let us consider the motion of p in latitude. For that pur 
pose let us resume the formula (Z ) of No. 514. If we neglect the pro 
duct of the inclinations by the excentricities of the orbits it will become 



the expression of R of No. 515 gives, in taking for the fixed plane that 
of the primitive orbit of p, 

/d R\ (* z (* z f r> t\\ - , i *. , f \ 

(~dj) ~ IT 5 """ ^ B cos< l ( " n + ^ > 

the value of i belonging to all whole positive and negative numbers in 
cluding also i = 0. Let 7 be the tangent of the inclination of the orbit 
of p , to the primitive orbit of ^, and n the longitude of the ascending 
node of the first of these orbits upon the second ; we shall have very 
nearly 

T! = a 7 sin. (n t + t ll) ; 
which gives 

= /- . 7 . sin. (n t + E n) ^ ^ B < l > y sin.(n t+en) 
- 



d z 



-. a! S B i 1 - 1 ) y sin. {i (n t n t + s s) + n t + s n] 



y 

the value here, as in what follows, extending to all whole positive and 
negative numbers, i = being alone excepted. The differential equation 



BOOK I.] NEWTON S PRINCIPIA. 129 

in 3 of will become, therefore, when the value of (-7- ) is multiplied by 
n 2 a 3 , which is equal to unity, 

= -j- t2 - + n 2 a u ft n 2 . -^ y sin. (n t + s 7 n) 

/ n 2 

H g a a B (i > y sin. (n t + g n) 

H r~ aa/2 B P-Dysin. {i (n t nt + s + nt+s n)] ; 



whence by integrating and observing that by 514 

8 s = a 3 u , 

s= - 7 sin. (n t + n) 




n t . y cos. (n t + -e n) 



: -n 2 -{n-i(n_nOF 7Sin ^ i(n/t "~ nt+/ ^ + nt + e - n ^ 
To find the latitude of p above a fixed plane a little inclined to that of 
its primitive orbit, by naming p the inclination of this orbit to the fixed 
plane, and 6 the longitude of its ascending node upon the same plane ; it 
will suffice to add to d s the quantity tan. p sin. (v 0), or tan. p sin. (n t 
+ 6 0, neglecting the excentricity of the orbit. Call p/ and ^ what p 
and 6 become relatively to (* . If ^ were in motion upon the primitive 

orbits of ft , the tangent of its latitude would be tan. p sin. (n t + s 6 } ; 

this tangent would be tan. p sin. (n t + e 6), if ft continued to move in 
its own primitive orbit. The difference of these two tangents is very 
nearly the tangent of the latitude of ft, above the plane of its primitive 
orbit, supposing it moved upon the primitive orbit of ft ; we have there 
fore 

tan. p sin. (n t + s _ ^) _ tan. p sin. (n t + s 0) = y sin. (n t + e n). 
Let 

tan. p sin. 6 = p ; tan. p sin. tf = p ; 
tan. p cos. 6 = q ; tan. p cos. tf = q ; 
we shall have 

y sin. n = p p ; y cos. n = q q 

and consequently if we denote by s the latitude of ft above the fixed plane, 
we shall very nearly have 
s = q sin. (n t + e) p cos. (n t + *) 

/& a 2 a 
4 (p p) B n t sin. (n t + ) 

Vor. II j 



130 A COMMENTARY ON [SECT. XI. 

m-|^ (q -q) B n t cos. (n t + ) 

^T^ * (q/ ~~ q) ^ (n/ t+l ] ~ (P/ "" P) C S (n/ + /) * 




519. Now let us recapitulate. Call (g) aud (v) the parts of the radius- 
vector and longitude v upon the orbit, which depend upon the elliptic 
motion, we shall have 

g = (g) + *S > v = (v) + 5v. 

The preceding value of s, will be the latitude of & above the fixed plane. 
But it will be more exact to employ, instead of its two first terms, which 
are independent of Xj the value of the latitude, which takes place in the 
case where p quits not the plane of its primitive orbit. These expressions 
contain all the theory of the planets, when we neglect the squares and the 
products of the excentricities and inclinations of the orbits, which is in 
most cases allowable. They moreover possess the advantage of being 
under a very simple form, and which shows the law of their different 
terms. 

Sometimes we shall have occasion to recur to terms depending on the 
squares and products of the excentricities and inclinations, and even to 
the superior powers and products. We can find these terms by the pre 
ceding analysis, the consideration which renders them necessary will al 
ways facilitate their determination. The approximations in which we 
must notice them, would introduce new terms which would depend upon 
new arguments. They would reproduce again the arguments, which the 
preceding approximations afford, but with coefficients still smaller and 
smaller, following that law which it is easy to perceive from the deve- 
lopement of R into a series, which was given in No. 515 ; an argument 
which, in the successive approximations, in found for thejirst time among the 
quantities of any order whatever r, and is reproduced only by quantities oj 
the orders r + 2, r-f-4, & c . 

Hence it follows that the coefficients of the terms of the form 

CITI 

t . . (n t + s), which enter into the expressions of g, v, and s, are ap- 
oos 

proximated up to quantities of the third order, that is to say, that the 
approximation in which we should have regard to the squares and pro- 



BOOK I.] NEWTON S PRINCIPIA. 131 

ducts of the excentricities and inclinations of the orbits would add nothing 
to their values ; they have therefore all the exactness that can be desired. 
This it is the more essential to observe, because the secular variations of 
the orbits depend upon these same coefficients. 

The several terms of the perturbations of g, v, s are comprised in the 
form 

sin 
k cos. *i (n t n t + e s) + r n t + r e} 9 

r being a whole positive number or zero, and k being a function of the 
excentricities and inclinations of the orbits of the order r, or of a superior 
order. Hence we may judge of what order is a term depending upon a 
given angle. 

It is evident that the motion of the bodies (* , (*/", &c. make it neces 
sary to add to the preceding values of g>, v, and s, terms analogous to 
those which result from the action of y! ; and that neglecting the square of 
the perturbing force, the sums of all these terms will give the whole va 
lues off, v and s. This follows from the nature of the formulas (X ), 
(Y), (Z ), which are linear relatively to quantities depending on the dis 
turbing force. 

Lastly, we shall have the perturbations of X, produced by the action of 
& by changing in the preceding formulas, a, n, h, 1, s, v, p, q, and (i! into 
a , n , H , 1 , s f , , p , q , and (i and reciprocally. 

THE SECULAR INEQUALITIES OF THE CELESTIAL MOTIONS. 

520. The perturbing forces of elliptical motion introduce into the expres- 

d v 

sions off, j-- , and s of the preceding Nos. the time t free from the sym 
bols sine and cosine, or under the form of arcs of a circle, which by in 
creasing indefinitely, must at length render the expressions defective. It 
is therefore essential to make these arcs disappear, and to obtain the 
functions which produce them by their developement into series. We 
have already given, for this purpose, a general method, from which it re 
sults that these arcs arise from the variations of elliptic motion, which are 
then functions of the time. These variations taking place very slowly 
have been denominated Secular Inequalities. Their theory is one of the 
most interesting subjects of the system of the world. We now proceed to 
expound it to the extent which its importance demands. 

1 2 



132 A COMMENTARY ON [SECT. XL 



By what has preceded we have 

1 h sin. (n t + ) 1 cos. (n t + t) &c. 



= a 



- U . C + 1 . V] . n t . sin. (n t + g) 

/ 

^-{h . C + h . D} . n t . cos. (n t + i) + y! S 



] 



d v 

-p = n + 2 n h sin. (nt + t) + 2 nl cos. (n t + t) + &c. 

Cl t 

^ {I C + T D] n 2 1 sin. (n t + ) 

+ ^ [h C + h D} n 2 t cos. (n t + i) + / T ; 

s = q sin. (n t + e) p cos. (n t + t) + &c. 

^- a 2 a (p p) B >. n t . sin. (n t + t) 

T* 

^- a 2 a (q q) B <. n t. cos. (n t + ) + ^ & > 

~k 

S, T, ^ being periodic functions of the time t. Consider first the expres 
sion of -j , and compare it with the expression of y in 510. The arbi 
trary n multiplying the arc t, under the periodic symbols, in the expres 
sion of jf- ; we ought then to make use of the following equations found 

( 1 L 

in No, 510, 

= X + 6.X." Y; 

= Y + 6 . Y" + X" 2 Z ; 
Let us see what these X, X , X", Y, &c. become. By comparing the ex 

pression of -3 - with that of y cited above, we find 

Cl L 

X = n + 2 n h sin. (n t + e) + 2 n 1 cos. (n t + s) + fjf T 

Y = (i! n 2 hC+h D} cos. (n t+t) ^ n * [I C + FD} sin. (nt+i). 

If we neglect the product of the partial differences of the constants by 

the perturbing masses, which is allowed, since these differences are of the 

order of the masses, we shall have by No. 510, 

X = (1^) U + 2 h sin. (n t + + 2 1 cos. (n t + 01 



+ 2 n (~) h cos. (nt + s) \ sin. (n t + )} 



+ 2 n()sin. (n t + ,) + 2 n()cos. (n t + ,); 
X" = 2 n ( ) [h cos. (n t + i) 1 sin. (n t + OJ 



BOOK I.] NEWTON S PRINCIPIA. 133 

The equation = X + 6 X" Y will thus become 

= (^ [I + 2 h sin. (n t + ) + 2 1 cos. (n t + i)J 

- - rl 1 

sin. (n t + t) + 2 n (^-J cos. (n t + t) 



it/n 2 h C + h D} cos. (nt + )4V n 2 Jl C+1 DJ sin.(n t + ). 
Equating separately to zero, the coefficients of like sines and cosines, we 
shall have 



If we integrate these equations, and if in their integrals we change 6 
into t, we shall have by No. 510, the values of the arbitrages in functions 
of t, and we shall be able to efface the circular arcs from the expressions 

d v 

of -; and of g. But instead of this change, we can immediately change 

01 I 

6 into t in these differential equations. The first of the equations shows 
us that n is constant, and since the arbitrary a of the expression for g de 

pends upon it, by reason of n 2 = 5, a is likewise constant. The two 

other equations do not suffice to determine h, 1, e. We shall have a new 

d v 
equation in observing that the expression of -= , gives, in integrating, 

(-1 t 

yn d t for the value of the mean longitude of p. But we have supposed 
this longitude equal to n t + s ; we therefore have n t+ = ,/n d t, which 
gives 

t 15 + ii - o- 

"dt + dt - 

and as we have T = 0, we have in like manner -j = 0. Thus the two 
d t d t 

arbitrages n and t are constants ; the arbitraries h, 1, will consequently be 
determined by means of the differential equations, 



2 

13 



134 A COMMENTARY ON [SECT. XI. 

The consideration of the expression of y^ having enabled us to deter 

mine the values of n, a, h, 1, and e, we perceive a priori., that the differen 
tial equations between the same quantities, which result from the expres 
sion of , ought to coincide with those preceding. This may easily be 
shown a posteriori, by applying to this expression the method of 510. 

Now let us consider the expression of s. Comparing it with that of y 
citetf above, we shall have 

X = q sin. (n t -f- e) p cos. (n t + ?) + & % 

Y = ^ . a* a B^ (p p ) sin. (n t + ) 

+ ^. a 2 a B> (q q ) cos. (n t + i), 
n and t, by what precedes, being constants; we shall have by No. 510, 



X" = 0. 
The equation = X + X" Y hence becomes 



= ) sin. (n t+ ) - cos. (n t + .) 
_ ^-% 2 a B (1 > (p p ) sin. (nt+ t) 

T? 

J? a 8 a B 1 ) (q q ) cos. (n t + ; 

TP 

whence we derive, by comparing the coefficients of the like sines and co 
sines, and changing d into t, in order to obtain directly p and q in 
functions of t, 

(q.-q ); (3) 



= .a*a<Ba>(P-P ); (4) 

When we shall have determined p and q by these equations, we shall 
substitute them in the preceding expression of s, effacing the terms which 
contain circular arcs, and we shall have 

s = q sin. (n t + s) p cos. (n t + t} + p %. 

521. The equation ~ = 0, found above, is one of great importance 

Cl L 

in the theory of the system of the world, inasmuch as it shows that the 
mean motions of the celestial bodies and the major-axes of their orbits are 
unalterable. But this equation is approximate to quantities of the order 



BOOK I.] NEWTON S PRINCIPIA. 135 

p h inclusively. If quantities of the order (j! h *, and following orders, 
produce in -v-^ , a term of the form 2 k t, k being a function of the ele 
ments of the orbits of ^ and yJ\ there will thence result in the expression of 
v, the term k t 2 , which by altering the longitude of p, proportionally to 
the time, must at length become extremely sensible. We shall then no 

longer have 

dn 

dl 
6ut instead of this equation we shall have by the preceding No. 

dn - 2k- 
dl 

It is therefore very important to know whether there are terms of the 
form k . t 2 in the expression of v. We now demonstrate, that if 
we retain only the Jirst power of the perturbing masses, however far may pro- 
ceed the approximation, relatively to the powers of the eccentricities and 
inclinations of the orbits, the expression v will not contain such terms. 

For this object we will resume the formula (X) of No. 513, 

acos.y/hdtf sin.v j 2/^R+gf-r ) r -asin.v/hdt.cos.v 

m V 1 e 2 

Let us consider that part of d g which contains the terms multiplied by t 2 , 
or for the greater generality, the terms which being multiplied by the sine 
or cosine of an angle a t + (3, in which a is very small, have at the same 
time a 2 for a divisor. It is clear that in supposing = 0, there will re 
sult a term multiplied by t 2 , so that the second case shall include the first. 
The terms which have the divisor a 2 , can evidently only result from a 
double integration ; they can only therefore be produced by that part of 
d g which contains the double integral signyi Examine first the term 
2 a cos. vfn d t (? sin. \fd R) 

m V (1 e 2 ) 
If we fix the origin of the angle v at the perihelion, we have 



1 + e cos. v 
and consequently 

a (1 _e 2 ) P 

COS. V = - " 1 ; 

9 f 

whence we derive by differentiating, 

a n e 1 ) 

p z d v . sin. v = i .dp; 

c 

14 



136 A COMMENTARY ON [SECT. XI. 

but we have, 

g* d v = d t V m a (1 e 2 ) = a 2 . n d t V I e s ; 
we shall, therefore, have 

a n d t g sin. v _ g d g 
V 1 e*~ ~e~ 
The term 

2 a cos, vy n d t . [g sin. vyV R] 

m VI e 2 
will therefore become 

R), or h y,z R _.. d R . 



It is evident, this last function, no longer containing double integrals, 
there cannot result from it any term having the divisor a 2 . 
Now let us consider the term 

_ 2 a si" v ./n d t [e cos. \fd R] 

m V 1 e* 

of the expression of d g. Substituting for cos. v, its preceding value in g, 
this term becomes 

2 asm, v/n d t. jg a (1 e*)} .fd R 

me V I e* 
We have 

g = aU+ie + ejfl, 

^ being an infinite series of cosines of the angle n t + i, arid of its multi 
ples ; we shall therefore have 

/E^J { g a(l -^}}fd R = a/n d t {% e + ^}fd R. 

Call %" the integral fyj n d t ; we shall have 
a/n d t . If e + %.}fd R = f a e/n d tfd R + a tf fd R a// . d R. 

These two last terms not containing a double integral sign, there can 
not thence result any term having a* for a divisor; reckoning only terms 
of this kind, we shall have 

2 a sin, v/n d t { cos. vfdE] __ 3 a* e sin, v/n d tfd R 
m V I e 2 m V I e z 



n d t m 
and the radius e will become 



.. . 
n d t/ m 



BOOK L] NEWTON S PRINCIPIA. 137 

(g) and ( -r:) being the expressions of P and of ~- , relative to the el 
liptic motion. Thus, to estimate in the expression of the radius-vector, 
that part of the perturbations, which is divided by 2 , it is sufficient to 

3 a 

augment by the quantity . x /n d t . fd R, the mean longitude 

n t + s, of this expression relative to the elliptic motion. 

Let us see how we ought to estimate this part of the perturbations in 
the expression of the longitude v. The formula (Y) of No. 516 gives by 

substituting ~ . ~-.fn d tfd R for d g and retaining only the terms 

111 II (.1 L 

divided by a 2 , 



a 2 n 2 d t * +1 J 3a 



v = v a "J: 1 ^ __ _ li. -"/n d tfd R; 

V 1 e* m 



But we have by what precedes 



j . . . , - 

d s = - ----- - ; fd T =r *n d t V 1 e 2 ; 

VI e 2 

whence it is easy to obtain, by substituting for cos. v its preceding value 
ing, 

2gd 2 g + dg 2 

a 2 n 2 d t 2 "*" d v 



V 1 e*~~ ~ nd t 

in estimating therefore only that part of the perturbations, which has the 
divisor a 2 , the longitude v will become 



(v) and ( p t ) being the parts of v and -i-p , relative to the elliptic mo 
tion. Thus, in order to estimate that part of the perturbations in the ex 
pression of the longitude of /A, we ought to follow the same rule which we 
have given with regard to the same in the expression of the radius-vector, 
that is to say, we must augment in the elliptic expression of the true 

longitude, the mean longitude n t + e by the quantity /n d tfd R. 

The constant part of the expression of (~ p.) developed into a series 

of cosines of the angle n t + and of its multiples, being reduced (see 
488) to unity, there thence results, in the expression of the longi- 



138 A COMMENTARY ON [SECT. XI. 

tude, the term f n d t / d R. If d R contain a constant term 
k y! . n cl t, this term will produce in the expression of the longitude v, 
the following one, =- . k n 2 1 2 . To ascertain the existence of such 

fd 111 

terms in this expression, we must therefore find whether d R contains a 
constant term. 

When the orbits are but little excentric and little inclined to one ano 
ther, we have seen, No. 518, that R can always be developed into an in 
finite series of sines and cosines of angles increasing proportionally to the 
time. We can represent them generally by the term 

k (if . cos. i n t + i n t + A}, 

i and i being whole positive or negative numbers or zero. The differen 
tial of this term, taken solely relatively to the mean motion of ^, is 

i k . y! . n d t . sin. {V n t + i n t + A}; 

this cannot be constant unless we have = i n + i n, which supposes 
the mean motions of the bodies //, and p to be parts of one another ; and 
since that does not take place in the solar system, we ought thence to con 
clude that the value of d R does not contain constant terms, and that in 
considering only the first power of the perturbing masses, the mean mo 
tions of the heavenly bodies, are uniform, or which comes to the same thing, 

^ = 0. The value of a being connected to n by means of the equation 

Cl L 

n z = j , it thence results that if we neglect the periodical quantities, the 

major-axes of the orbits are constant. 

If the mean motions of the bodies ^ and /u/ 9 without being exactly com 
mensurable, approach, however, very nearly to that condition, there will 
exist in the theory of their motions, inequalities of a long period, and 
which, by reason of the smallness of the divisor 2 , will become very sen 
sible. We shall see hereafter this is the case with regard to Jupiter and- 
Saturn. The preceding analysis will give, in a very simple manner, that 
part of the perturbations which depend upon this divisor. It hence re 
sults that in this case it is sufficient to vary the mean longitude n t + f 

3 a 

ory*n d t by the quantity fn d tfd R; or, which is the same, to aug 
ment n in the integral,/ n d t by the quantity - ~fd Rj but consider- 



BOOK L] NEWTON S PRINCIPIA. 



m 



ing the orbit of ^ as a variable ellipse, we have n e = 3 ; the preceding 

variation of n introduces, therefore, in the semi-axis-major a of the orbit, 

2 a*fd R 

the variation * . 

If we carry the approximation of the value -r , to quantities of the 

vl L 

order of the squares of the perturbing masses, we shall find terms propor 
tional to the time ; but considering attentively the differential equations of 
the motion of the bodies /A, /& , &c. we shall easily perceive that these terms 
are at the same time of the order of the squares and products of the ex- 
centricities and inclinations of the orbits. Since, however, every thing 
which affects the mean motion, may at length become very sensible, we 
shall now notice these terms, and perceive that they produce the secular 
equations observed in the motion of the moon. 

522. Let us resume the equations (1) and (2) of No. 520, and suppose 
_(i. . n . C m ,-Q-JT _ /a .n.D 

they will become 

Vl " t r*. V 1 fTI VI 1 / 



dl 



The expression of (0, 1) and of |0, 1| may be very simply determined in 
this way. Substituting, instead of C and D, their values determined in 
No. 517, we shall have 



We have by No. 516, 

db d 2 b (0 > 



d b (0 > d s b <> 
and we shall easily obtain, by the same No. 5 and . 2 in functions 

Q CX, tl 06 

of b (0) and b (1) ; and these quantities are given in linear functions of b (0) 

* i -4 



140 A COMMENTARY ON [SECT. XL 

and of b (1) ; this being done, we shall find 

~~ 2 

3 a 2 b (1 > 

f d_A^ ,d A\ -I . 

V da ; V da 2 ) -2(1 *) 2 

wherefore 

3 ^. n . 2 . b U 

cu) = - M i_.r 

Let 

(a 2 2 a a cos. 6 + a z )*= (a, a ) + (a, a ) cos. 0+(a, a ) 7 cos. 
we shall have by No. 516. 

(a, a ) = a , b <> ; (a, a ) = a , b >, &c. 



We shall, therefore, have 
_ 

Next we have, by 516, 



_ 8g . na*a . (a, a ) 

4 (a 2 a 2 ) 2 



d b (1 > d 2 b W 



.-j -- . -j- . 
da da 2 j 

2 

Substituting for b (1) and its differences, their values in b (0) and b (1) , we 

I -* -k 

shall find the preceding function equal to 

Q f(l+a 2 )b) + Jb)l 

8a l -i -U 



therefore 

+ a 2 ) 



2 (1 a 2 ) 2 
or 

. rr 3 /& . a n(a 2 + a 2 ) (a, a ) + a a (a, a }} 

I^JJ - 2 (a 2 a 2 ) 2 

We shall, therefore, thus obtain very simple expressions of (0, 1) and 

of JO, 1|, and it is easy to perceive from the values in the series of b (0) and 

i 

~~ e 

of b (1) , given in the No. 516, that these expressions are positive,- if n is 

~ 2 

positive, and negative if n is negative. 

Call (0, 2) and |0, 2|, what (0, 1) and |0, 1| become, when we change a 



BOOK I.] 



NEWTON S PRINCIPIA. 



141 



and i/f into a" and &". In like manner let (0, 3), and (0, 3) be what the 
same quantities become, when we change a and p f into a!" and //" ; and 
so on. Moreover let h", 1" ; h" , 1 ", &c. denote the values of h and 1 
relative to the bodies / , ft ", &c. Then, in virtue of the united actions of 
the different bodies /* , /<// , p " 9 &c. upon p, we shall have 

^i ={(o, i) + (o, 2) + (o, 3) + &c.ji [oTi|.r [M.!" &c. ; 

(.1 L 



_, 

Cl L 

It is evident that , 



; &c. will be determined by 



expressions similar to those of T - and of ^; and they are easily obtain 

ed by changing successively what is relative to & into that which relates 
to //, , // , &c. and reciprocally. Let therefore 

(1,0), IM)]; (1,2), O; &c. 
be what 

; &c. 



(0,1), JOTT); (0,2), | 
become, when we change that which is relative to , into what is relative 
to p and reciprocally. Let moreover 

(2,0), gof; ( 2 >!)> 153? &c - 

be what 

(0,2), IM; (0, 1), |M ; &c. 

become, when we change what is relative to ^ into what is relative to /." 
and reciprocally; and so on. The preceding differential equations re 
ferred successively to the bodies /-, /"- , ,<*", &c. will give for determining 
h, 1, h , 1 , h", 1", &c. the following system of equations, 

= {(0, 1) + (0, 2) + &c.] 1 |0, 1|. 1 [0, 2| 1" &c. 
-! = -J(0,l)+(0,2) + &c.]h- 



|0,2|b"+&c. 



dh 



= f(l, 0) + (1, 2) + &c.$l 11, Oj. 1 [M 1" &c. 
11 ={(1,0) + (l,2) + &c.lh +[I7o|.h + 

Cl L 

1 h - = {(2, 0) + (2, 1) + &C.J1" g] 



- |M. T - &c. 



dl r/ 
dt 
&c. 



= {(-2, 0) + (2, 1) + &c.} . h" + [2,01 h + 12, 1| h + &c. 



(A) 



142 A COMMENTARY ON [SECT. XI. 



The quantities (0, 1) and (1, 0), |0, 1) and |I, 0| have remarkable rela 
tions, which facilitate the operations, and will be useful hereafter. By 
what precedes we have 

(0, 1) = - . / r g jpy-j . 

If in this expression of (0, 1) we change /// into /w, n into n , a into a 
and reciprocally, we shall have the expression of (1, 0), which will con 
sequently be 

, - SAt.n a 8 . a (a/ a/ 

4 (a 2 a 2 ) 2 ; 

but we have (a, a / = (a , a) , since both these quantities result from th 
developement of the function (a 2 2 a a cos 6 + a 2 ) s into a series or 
dered according to the cosine of 6 and of its multiples. We shall, there 
fore, have 

(0, 1). ^ n a = (1, 0). ft. , n a. 
But, neglecting the masses /a, /, , &c. in comparison ,with M, we have 

M /2 M 
n 2 = . -,\ rr 2 = -j-.\ &c. 
a 3 a 3 

Therefore 

(0, 1) ft V a = (I, 0) ft! V a ; 

an equation from which we easily derive (1, 0) when (0, 1) is determined. 
In the same manner we shall find, 

|0, 1| ft V a = |J70| tf V a . 

These two equations will also subsist in the case where n and n have 
different signs ; that is to say, if the two bodies /*, /* circulated in different 
directions ; but then we must give the sign of n to the radical V a, and 
the sign of n to the radical V a . 

From the two preceding equations evidently result these 
(0, 2) fj, V a = (2, 0) ft" V a."-, \0^2\ ft V a = [2J"o|. p" V a" , &c. 
(I, 2) (if V a = (2, 1) (jJ 1 V a"; [\^2\ p V a = gjj. p" V a"; &c. 
523. To integrate the equations (A) of the preceding No., we shall 
make 

h = N. sin. (g t + /3) ; 1 = N . cos. (g t + /3) ; 
h = N . sin. (g t + /3) ; 1 = N cos. (g t + /3) ; 
&c. 

Then substituting these values in the equations (A), we shall have 
N g ={(0, 1) + (0, 2) + &c.JN IM]. N " 
N g=KM) + 2) + &c.]N jTToJ. N 
N"g = J(2, 0) + (2, 1) + c.}N" [270|. N 



Ml N &cj 



BOOK I.] NEWTON S PRINCIPIA. 143 

If we suppose the number of the bodies /, //, X > &c. equal to i ; these 
equations will be in number i, and eliminating from them the constants 
N, N , &c., we shall have a final equation in g, of the degree i, which we 
easily obtain as follows : 
Let <p be the function 

N 2 . ^ V a {g (0, 1) (0, 2) &c.} 
+ N V V a {g (I, 0) (1, 2) &c.} 
+ &c. 



+ 2 N ft V a i|o7Tj N + |OT2| N" + &c.J 
+ 2 N> V a l|TT2[N" + jl73| N "+ &c.} 
+ 2 N>" V a" J|2[3] N " + &c.} 

+ &c. 

The equations (B) are reducible from the relations given in the pre 
ceding No. to these 

- o- &c 





Considering therefore, N, N , N", &c. as so many variables, <f> will be 
a maximum. Moreover, <p being a homogeneous function of these varia 
bles, of the second dimension ; we have 



we have, therefore, <p = 0, in virtue of the preceding equations. 

Thus we can determine the maximum of the function <p. We shall first 
differentiate this function relatively to N, and then substitute in p, for N, 

its value derived from the equation (-pcf) = 0, a value which will be a 

linear function of the quantities N r , N", &c. In this manner we shall 
have a rational function whole and homogeneous of the second dimension 
in terms of N r , N", &c. : let <p (1) be this function. We shall differentiate 
<f> (1) relatively to N , and we shall substitute in <p (1) for N r its value derived 

from the equation -cr = : we shall have a homogeneous function 



of the second dimension in N", N" , &c. : let <p (2) be this function. Con 
tinuing thus, we shall arrive at a function <p (i ~ J) of the second dimension, 
in N Ci ~ ]) and which will consequently be of the form (N (i ~ V) z . k, k being 
a function of g and constants. If we equal to zero, the differential of 
<p (i ~ 1 ) taken relatively to N^" 1 , we shall have k = 0; which will give 
an equation in g of the degree i, and whose different roots will give as 
many different systems for the indeterminates N, N , N", &c. : the inde- 



144 A COMMENTARY ON [SECT. XL 

terminate N^- 1 ) will be the arbitrary of each system; and we shall im 
mediately obtain, the relation of the other indeterminates N, N , &c. of 
the same system, to this one, by means of the preceding equations taken 
in an inverse order, viz., 

p<i-3)x 
^- 3 



Let g, gi, g 2 , &c. be the i roots of the equation in g : let N, N , N", &c. 
be the system of indeterminates, relative to the rootg: letN /5 N/, N/ , &c. 
be the system of indeterminates relative to the root g b and so on : by the 
known theory of linear differential equations, we shall have 
h = N sin. (g t + /3) + N! sin. (g, t + ft) + N 2 (g 8 t + &) + &c. ; 
h = N sin. (g t + /3) + N/ sin. ( gl t + ft) + N 2 (g 2 t + &) + &c. ; 
h"= N"sin. (g t + /3) + N/ sin. ( gl t + ft) + N 2 "(g. 2 t + &) + &c. ; 

&c. 

ft ftj ft} & c being arbitrary constants. Changing in these values of 
h, h , h", &c. the sines into cosines ; we shall have the values of 1, 1 , 1", &c. 
These different values contain twice as many arbitraries as there are roots 
g, g l5 g 2 , &c. ; for each system of indeterminates contains an arbitrary, 
and moreover, it has i arbitraries /3, ft, /3 2 , &c. ; these values are conse 
quently the complete integrals of the equations (A) of the preceding 
No. 

It is necessary, however, to determine only the constants N, N 1} &c. ; 
N, N/, &c. ; ft ft, &c. Observations will not give immediately the con 
stants, but they make known at a given epoch, the excentricities e, e , &c. 
of the orbits, and the longitudes , ?/, &c. of their perihelions, and conse 
quently, the values of h, h , &c., 1, 1 , &c. : we shall thus derive the values 
of the preceding constants. For that purpose, we shall observe that if 
we multiply the first, third, fifth, &c. of the differentia] equations (A) of 
the preceding No., respectively by N. /*. V a, N . /a . V a , &c. ; we 
shall have in virtue of equations (B), and the relations found in the pre 
ceding No. between (0, 1) and (1, 0), (0, 2), and (2, 0), &c. 

N . ~ ft, V a + N . i^ m V a + N". ^~ y! V a" + &c. 

= g {N. 1 . ft. V a + N . 1 . i* . V a + N". 1". ft". V a" + &c.} 
If we substitute in this equation for h, h , &c. 1, 1 , &c. their preceding 
values ; we shall have by comparing the coefficients of the same cosines 
= N . Nj . ft V a + N . NI . & V a + N". N,". ft". V a" + &c. ; 
= N . N 2 . ft V a + N . Ng . p V a + N". N 2 ". ft". V a" + &c. 



BOOK L] NEWTON S PRINCIPLE 145 

Again, if we multiply the preceding values of h, h , &c. respectively by 

N./tt. V a, W.fif. V a , &c. 
we shall have, in virtue of these last equations, 
N . ft h . V a + N (i/. h . V a + N". // h". V a" + &c. 

= {N 2 . p . V a + N". {* . V a! + N" 2 . p". V a" + &c.} sin (g t + |8) 

In like manner, we have 

N . p 1 . V a + N . ^ 1 . V a + N". // 1". V a" + &c. 
= N 2 . ^ . V a + N 2 . ^. V a + N" 2 . ^. V a" + &c.J cos. (g t + /3). 

By fixing the origin of the time t at the epoch for which the values of 
h, 1, h , 1 , &c. are supposed known ; the two preceding equations give 

tan B = N h ** V a + N/ h/ <* V a/ + N " h " A*" ^ a " + &c - 
~ N . 1 p . V a + N . 1 V. V of + N". I" // . V a" + &c. * 

This expression of tan. /3 contains no indeterminate ; for although the 
constants N, N , N", &c. depend upon the indeterminate N (i ~ l \ yet, as 
their relations to this indeterminate are known by what precedes, it will 
disappear from the expression of tan. B. Having thus determined /3, we 
shall have N (i ~ l \ by means of one of the two equations which give tan. /3; 
and we thence obtain the system of indeterminates, N, N , N", &c. rela 
tive to the root g. Changing, in the preceding expressions, this root into 
gi &25 gsj &c. we shall have the values of the arbitraries relative to each 
of these roots. 

If we substitute these values in the expressions of h, 1, h , 1 , &c. ; we 
hence derive the values of the excentricities e, e , &c. of the orbits, and 
the longitudes of their perihelions, by means of the equations 

e 2 = h 2 + l 2 ; e /2 = h /2 + 1 /2 ; &c. 

h h 

tan. nt = -j- ; tan. / = p- ; &c. 

we shall thus have 

e 2 = N 2 + Ni 2 + N 2 2 + & c . + 2 N N; cos. J( gl g) t + ft ] 

+ 2 N N 2 cos. J( gsr - g ) t + &-<3) } + 2 N! N 2 cos.{( Sf -gi) t+^-ft} +&c. 

This quantity is always less than (N + N! + N 2 + &c.) 2 , when the 
roots g, gl , &c. are all real and unequal, by taking positively the quanti 
ties N, NI, &c. In like manner, we shall have 

tan * = N sin< (g f + ^) + N I sin - (gi t + ft) + N 2 sin. (g 2 1 + &) + &c. 
N cos. (g t + /3) + N, cos. ( gl t + ft) + N 2 cos. (g- 2 1 + &) + &c. 

whence it is easy to get, 

tan ( w _cr t _S)= N I sin - Ugi-g) t + ff.-ffj + N 2 sin. 
1 



146 A COMMENTARY ON [SECT. XL 

Whilst the sum NI + N 2 + &c. of the coefficients of the cosines of 
the denominator, all taken positively, is less than N, tan. (a g t 0) 
can never become infinite ; the angle g t /3 can never reach the 
quarter of the circumference ; so that in this case the true mean motion 
of the perihelion is equal to g t. 

524. From what has been shown it follows, that the excentricities of 
the orbits and the positions of their axis-majors, are subject to considera 
ble variations, which at length change the nature of the orbits, and whose 
periods depending on the roots g, g 1? g 2 , &c., embrace with regard to the 
planets, a great number of ages. We may thus consider the excentrici 
ties as variably elliptic, and the motions of the perihelions as not uniform. 
These variations are very sensible in the satellites of Jupiter, and we shall 
see hereafter, that they explain the singular inequalities, observed in the 
motion of the third satellite. 

But it is of importance to examine whether the variations of the excen 
tricities have limits, and whether the orbits are constantly almost circular. 
We know that if the roots of the equation in g are all real and unequal, 
the excentricity e of the orbit of p is always less than the sum N + NI 
+ N 2 + & c - f the coefficients of the sines of the expression of h taken 
positively ; and since the coefficients are supposed very small, the value 
of e will always be inconsiderable. By taking notice, therefore, of the 
secular variations only, we see that the orbits of the bodies /A, /& , /*", &c. 
will only flatten more or less in departing a little from the circular form ; 
but the positions of their axis-majors will undergo considerable variations. 
These axes will be constantly of the same length, and the mean motions 
which depend upon them will always be uniform, as we have seen in No. 
521. The preceding results, founded upon the smallness of the excentricity 
of the orbits, will subsist without ceasing, and will extend to all ages past 
and future ; so that we may affirm that at any time, the orbits of the 
planets and satellites have never been nor ever will be very excentric, at 
least whilst we only consider their mutual actions. But it would not be 
the same if any of the roots g, g l5 g 2 , &c. were equal or imaginary : the 
sines and cosines of the expressions of h, 1, h , 1 , &c. corresponding to 
these roots, would then change into circular arcs or exponentials, and 
since these quantities increase indefinitely with the time, the orbits would 
at length become very excentric ; the stability of the planetary system 
would then be destroyed, and the results found above would cease to 
take place. It is therefore highly important to show that g, gi, gg, &c. 
are all real and unequal. This we will now demonstrate in a very simple 



BOOK I.] NEWTON S PRINCIPIA. 147 

manner, for the case of nature, in which the bodies ^ ,. , &", &c. of the 
system, all circulate in the same direction. 

Let us resume the equations (A) of No. 528. If we multiply the first 
by p . V a . h ; the second, by /A . V a . 1 ; the third by /u/. V a , h ; the 
fourth by /* . V a . 1 , &c. and afterwards add the results together ; the 
coefficients of h 1, h 1 , h" 1", &c. will be nothing in this sum, the coeffi 
cients of h 1 h 1 will be |07T|. p . V a flTO). ^ . V a , and this will 
be nothing in virtue of the equation |0, 1|. ft. V a = |1, 0|. //. V a found 
in No. 522. The coefficients of h" 1 h 1", h" 1 h 1", &c. will be 
nothing for the same reason ; the sum of the equations (A) thus prepared 
will therefore be reduced to 

hdh + ldl , h dh + l dl , . , , 

^ .p.V a + gi .(* . Va + &c. = 0; 

and consequently to 

= e d e . ft, . V a + e d e . p!. V a + &c. 

Integrating this equation and observing that (No. 52-1) the semi-axis- 
majors are constant, we shall have 

e z . (t V a + e 2 . //. V a + e" 2 . fil . V a" + &c. = constant ; (a) 

The bodies /A, ^ , fj/ 9 &c. however being supposed to circulate in the 
same direction, the radicals V a, V a , V a", &c. ought to be taken po 
sitively in the preceding equation, as we have seen in No. 522; all the 
terms of the first member of this equation are therefore positive, and con 
sequently, each of them is less than the constant of the second member. 
But by supposing at any epoch the excentricities to be very small, this 
constant will be very small ; each of the terms of the equation will, there 
fore, remain always very small and cannot increase indefinitely ; the orbits 
will always be very nearly circular. 

The case which we have thus examined, is that of the planets and 
satellites of the solar system ; since all these bodies circulate in the same 
direction, and at the present epoch their orbits have little excentricity. 
That no doubt may exist as to a result so important, we shall observe 
that if the equation which determines g, contained imaginary roots, some 
of the sines and cosines of the expressions of h, 1, h , 1 , &c. would trans 
form into exponentials ; thus the expression of h would contain a finite 
number of terms of the form P . c ft , c being the number of which the 
hyperbolic logarithm is unity, and P being a real quantity, because h or 
e sin. w is a real quantity. Let 

Q.cf<,P . c f<, Q .cf<,P".cf ,&c. 

be the corresponding terms of 1, h , 1 , h", &c. ; Q, P , Q , P", & c . being 

K2 



H8 A COMMENTARY ON [SECT. XI. 

also real quantities : the expression of e 2 will contain the term (P 2 + Q c ) 
c zft ; the expression of e 2 will contain the term (P 2 + Q 2 ) c 2 f c , and 
so on ; the first member of the equation (u) will therefore contain the 
term 



If v therefore, we suppose c f c to be the greatest of the exponentials 
which contain h, 1, h , 1 , &c. that is to say, that in which f is the most 
considerable, c 2ft will be the greatest of the exponentials which contain 
the first member of the preceding equation : the preceding term cannot 
therefore be destroyed by any other term of this first member ; so that for 
this member to be reduced to a constant, the coefficient of c 2ft must be 
nothing, which gives 

=(P 2 +Q ~)^ Va + (P 2 +Q 2 ) / /v a + (P" 2 + Q"V Va" + &c. 
When V a, V a , V a", &c. have the same sign, or which is tantamount, 
when the bodies /ct, /, , /,", &c. circulate in the same direction, this equa 
tion is impossible, provided we do not suppose P = 0, Q = 0, P = 0, &c.; 
whence it follows that the quantities h, 1, h ] , &c. do not contain expo 
nentials, and that the equation in g does not contain imaginary roots. 

If this equation had equal roots, the expressions of h, 1, h , 1 , &c. would 
contain as we know, circular arcs and in the expression of h, we should 
have a finite number of terms of the form P t r . Let Q t r , P t r , Q t r , &c. 
be the corresponding terms of 1, h , 1 , &c. P, Q, P , Q , &c. being real 
quantities; the first member of the equation (u) will contain the term 
{(P z + Q~)f*V a + (P 2 +Q /2 ) /a V a + (P" 2 + Q" 2 ) p." V a" + &c.}. t 2r . 

If t r is the highest power of t, contained by the values of h, 1, h V, &c. ; 
t 2 r will be the highest power of t contained in the first member of the 
equation (u) ; thus, that this member may be reduced to a constant, we 
must have 

= (P 2 +Q 2 )/* Va + (P /2 + Q )/<* Va + &c. 
which gives 

P = 0, Q = 0, P = 0, Q = 0, &c. 

The expressions of h, 1, h , T, &c. contain therefore, neither exponen 
tials nor circular arcs, and consequently all the roots of the equation in g 
are real and unequal. 

The system of the orbits of /., ///, // , &c. is therefore perfectly stable 
relatively to their excentricities ; these orbits merely oscillate about a 
mean state of ellipticity, which they depart from but little by preserving 
the same major-axis : their excentricities are always subject to this condi- 



BOOK I.] 



NEWTON S PR1NCIPIA. 



149 



tion, viz. that the sum of their squares multiplied respectively by the masses 
of the bodies and by the square roots of the major-axes is always the same. 

525. When we shall have determined, by what precedes, the values of 
e and of ; we shall substitute in all the terms of the expressions of f ; 

and -T- , given in the preceding Nos., effacing the terms which contain 

Cl I 

the time t without the symbols sine and cosine. The elliptic part of these 
expressions will be the same as in the case of an orbit not disturbed, with 
this only difference, that the excentricity and the position of the perihe 
lion are variable ; but the periods of these variations being very long, by 
reason of the smallness of the masses ^, v> , /*", &c. relatively to M ; we 
may suppose these variations proportional to the time, during a great 
interval, which, for the planets, may extend to many ages before and 
after the given epoch. 

It is useful, for astronomical purposes, to obtain under this form, the 
secular variations of the excentricities and perihelions of the orbits : we 
may easily get them from the preceding formulae. In fact, the equation 
e 2 = h * + 1 2 , gives ederrhdh+ldl; but in considering only the 
action of //, we have by No. 522, 



wherefore 



h I J; 



but we have h 1 h 1 = e e sin. (/ ^) ; we, therefore, have 

.) ; 



. e sn. 



thus, with regard to the reciprocal action of the different bodies /* , ,.", &c. 
we shall have 






- |oTl|. e sin. (~ ) + [072|. e" sin. (*" ) + & 

^ - . 



&c. 



d e 



- = ]I70J e sin. ( ) + |1, 2| e" sin. (" ) + &c. 

t "~ ^ 



d t 
&c. 



e sn. 



2~J e sin. 



&c. 



K3 



150 A COMMENTARY ON [SECT. XL 

The equation tan. = y , gives by differentiating 

e 2 d w = 1 d h hdl. 

With respect only to the action of p, , by substituting for d h and d 1 
their values, we shall have 



= (0, 1) (h* + P) _ JOTTJ. {h h + 1 1 }; 



which gives 



^ = (0, 1) |0, 1[. X COS. (J w); 

we shall, therefore, have, through the reciprocal actions of the bodies 

ft, (jf, fjf t &C. 

^ =(0,l) + (0,2) + &c. (OH]. cos.( w ) |(jr2]. cos.(w" *} &c. 



d^ =( 

f^i< 

&c. 

If we multiply these values of -r- , -5 , &c. T- , T- , &c. by the time t ; 

we shall have the differential expressions of the secular variations of the 
excentricities and of the perihelions, and these expressions which are only 
rigorous whilst t is indefinitely small, will however serve for a long in 
terval relatively to the planets. Their comparison with precise and distant 
observations, affords the most exact mode of determining the masses of the 
planets which have no satellites. For any time t we have the excentricity 
e, equal to 

de t 2 d z e 

e, -i , -T i , &c. being relative to the origin of the time t or to the given 

Cl t Cl L 

d e 
epoch. The preceding value of -5 will give, by differentiating it, and 

d * e d 3 e 
observing that a, a , &c. are constant, the values of -75 , -7-7-3) &c. ; we 

Cl L Cl I 

can, therefore, thus continue as far as we wish, the preceding series, and 
by the same process, the series also relative to -a : but relatively to the 
planets, it will be sufficient, in comparing the most ancient observations 



BOOK I.] 



NEWTON S PRINCIPIA. 



151 



which have come down to us, to take into account the square of the time, 
in the expressions of the series of e, e , &c. -a, /, &c. 

526. We will now consider the equations relative to the position of the 
orbits. For this purpose let us resume the equations (3) and (4) of 

No. 520, 



- 



By No. 5 16, we have 
a 2 a . 

and by the same No., 



We shall therefore have 



= a. 



Sb> 



\ 2 



3 tif . n . a 2 b (i) 



4 4 (1 a 2 ) 2 

The second member of this equation is what we have denoted by (0, 1) 
in 522 ; we shall hence have 

^| = (0, 1) (q - q) ; 

^ = (0,l)(p-p ); 

Hence, it is easy to conclude that the values of q, p, q , p , &c. will be 
determined by the following system of differential equations : 

j-3 = J(0, 1) + (0,2) +&c.}. p (0, l)p (0, 2)p" &c. 

^=-uo, 



= { ( l > 0) 



-$-=: (2 ; 0) + (2, 

&c. 



2 )+&c.} . q + (0, 1) q + (0, 2) q" + &c. 



, 2)+&c.J . q + (1, 0) q + (1, 2) q" + &c. 
+ (2,1) +&c.}.p" (2,0)p (2, l)p _& c . 
\ -q" +(2,0)q+(2, l)q + &c. 

K 4 



(0 



153 A COMMENTARY ON [SECT. XI. 

This system of equations is similar to that of the equations (A) of No. 
522: it would entirely coincide with it, if in the equations of (A) we were 
to change h, 1, h , F, &c. into q, p, q , p , &c. and if we were to suppose 

|OTT| = (0,1); 



|lQ| = (1,0); 

&c. 

Hence, the process which we have used in No. 528 to integrate the 
equations (A) applies also to the equations (C). We shall therefore 
suppose 

q =N Gos.(gt+/3) + N lC os. (git+/S,) + N 8 cos. ( 

p =N sin. (gt+/3) + N! sin. ( gl t + /3 1 ) + N 2 sin. ( 

q = N cos. (gt+/3) + N/cos. (git+ft)+N 2 cos.(g 2 t+/3 2 )+&c. 

p = N sin. (gt+^ + N/sin. ( gl t+ft) + N 2 sin. (g a t+&) + &c. 

&c. 

and by No. 523, we shall have an equation in g of the degree i, and whose 
different roots will be g, g l9 g 2 , &c. It is easy to perceive that one of 
these roots is nothing; for it is clear we satisfy the equations (C) by sup 
posing p, p , p", &c. equal and constant, as also q, q , q", & c . This 
requires one of the roots of the equation in g to be zero, and we can 
thence depress the equation to the degree i ]. The arbitraries 
N, Nj, N , &c. /3, /3 15 &c. will be determined by the method exposed in 
No. 523. Finally, we shall find by the process employed in No. 524. 

const. = (p 2 + q 2 ) p V a + (p /2 + q 2 ) tf V a + &c. 
Whence we conclude, as in the No. cited, that the expressions of p, q, 
p , q 7 , &c. contain neither circular arcs nor exponentials, when the bodies 
p, yJ> p", &c. circulate in the same direction : and that therefore the equa 
tion in g has all its roots real and unequal. 

We may obtain two other integrals of the equations (C). In fact, if 
we multiply the first of these equations by /M V a, the third by /// V a , 
the fifth by // V a", &c. we shall have, because of the relations found in 
No. 522, 

= 3~t " V a + Tt a/ V H/ + &c>; 
which by integration gives 

constant = q ^ V a + q /". V of + &c (1) 

In the same manner we find 

constant = p ^ V a + p /// V of + &c. . . . . (2) 
Call <p the inclinatior of the orbit of p to the fixed plane, and 6 the Ion- 



BOOK I.] NEWTON S PRINCIPIA. 153 

gitude of the ascending node of this orbit upon the same plane ; the lati 
tude of i* will be very nearly tan. <p sin. (n t -f ^) : Comparing this 
value with q sin. (n t + t) p cos. (n t + we shall have 

p = tan. <p sin. d ; q = tan. <p cos. d ; 
whence we obtain 

tan. <p = V (p 2 + q 2 ) ; tan. d = - ; 

We shall, therefore, have the inclination of the orbit of //, and tne po 
sition of its node, by means of the values of p and q. By marking suc 
cessively with one dash, two dashes, &c. relatively to /M/, /", &c. the values 
of tan. <p, tan. 0, we shall have the inclinations of the orbits of // p", &c. 
and the positions of their nodes by means of p , q , p", q", &c. 

The quantity V p 2 + q 2 is less than the sum N -j- Nj + N 2 + &c. of 
the coefficients of the sines in the expression of q ; thus, the coefficients 
being very small since the orbit is supposed but little inclined to the fixed 
plane, its inclination will always be inconsiderable ; whence it follows, that 
the system of orbits is also stable, relatively to their inclinations as also to 
their excentricities. We may therefore consider the inclinations of the 
orbits, as variable quantities comprised within determinate limits, and the 
motion of the nodes as not uniform. These variations are very sensible 
in the satellites of Jupiter, and we shall see hereafter, that they explain 
the singular phenomena observed in the inclination of the orbit of the 
fourth satellite. 

From the preceding expressions of p and q results this theorem : 

Let us imagine a circle whose inclination to a fixed plane is N, and of 
which the longitude of the ascending node is g t + ft ; a ^ so ^ us imagine 
upon this first circle, a second circle inclined by the angle NI , the longitude 
of whose intersection with the former circle is gi t + ft ; upon this second 
circle let there be a third inclined to it by the angle N 2 , the longitude of 
whose intersection with the second circle is g 2 t + j3 2 , and so on ; the po 
sition of the last circle will be that of the orbit of p. 

Applying the same construction to the expressions of h and 1 of No. 
523, we see that the tangent of the inclination of the last circle upon the 
fixed plane, is equal to the excentricity of /* s orbit, and that the longitude 
of the intersection of this circle with the same plane, is equal to that of 
the perihelion of /t s orbit. 

527. It is useful for astronomical purposes, to have the differential va 
riations of the nodes and inclinations of the orbits. For this purpose, let 
us resume the equations of the preceding No. 



154 A COMMENTARY ON [SECT. XL 

tan. ? = V (p 2 + q 2 ), tan. 6 = - . 

Differentiating these, we shall have 

d if) = d p sin. 6 + d q cos. 6 ; 
, _ d p cos. 6 d q sin. 6 

tan. p 

If we substitute for d p and d q, their values given by the equations (C) 
of the preceding No. we shall have 



j= (0, 1) tan. <f> sin. (6 ff) + (0, 2) tan p" . sin. (d 
^=_ {(0, l)+(0,2) + &c.J+(0, 1) cos . ( , _ , 




In like manner, we shall have 

-^=(1, 0) tan. <p sin. (6f f) + (\, 2) tan. 9" sin. (* 0" 

(1 L 



&c. 

Astronomers refer the celestial motions to the moveable orbit of the 
earth ; it is in fact from the plane of this orbit that we observe them ; it is 
therefore important to know the variations of the nodes and the inclina 
tions of the orbits, relatively to the orbit of one of the bodies /*, p , /A", &c. 
for example to the orbit of /z. It is clear that 

q sin. (n t + ?} p cos. (n t + f) 

would be the latitude of ft, above the fixed plane if it were in motion upon 
the orbit of p. The latitude of this moveable plane above the same 
plane is 

q sin. (n t + e) p cos. (n t + e ) 

but the difference of these two latitudes is very nearly the latitude of ftf 
above the orbit of p; calling therefore <p/ the inclination, and dj the lon 
gitude of the node of / upon the orbit of ft, we shall have, by what 
precedes, 



tan. p/ = V (p _p)+ (q q) 2 ; tan. */ = jr 
If we take for the fixed plane, that of (Ss orbit at a given epoch ; we 



BOOK I.] NEWTON S PRINCIPIA. 155 

shall have at that epoch p = 0, q = ; but the differentials d p and d q 
will not be zero ; thus we shall have. 

d p; = (dp dp) sin. 8 + (d q d q) cos. ff ; 

d P d p) cos. 8 (d q d q) sin. & 
tan. <p 

Substituting for d p, d q, d p , d q , &c. their values given by the equa 
tions (C) of the preceding No., we shall have 

ijjL = (1, 2) (0, 2)} tan. p" sin. (ff 6") 

+ {(i 9 3) __ (0, 3)} tan. ? " sin. (ff *" ) + &c. 
- = f (1, 0) + (1, 2) + (1, 3) + &c.J (0, 1) 

-j. (i, g) (0, 2)] . ~ -^ cos. (ff ff ) 

+ {(I, 3) (0, 3)] . -^ Bl__ cos. ff ff") + &c. 

It is easy to obtain from these expressions the variations of the nodes, 
and inclinations of the orbits of the other bodies (*" 9 ^ "^ &c. upon the 
moveable orbit of p. 

528. The integrals found above, of the differential equations which deter 
mine the variations of the elements of the orbits, are only approximate, and 
the relations which they give among the elements, only take place on the 
supposition that the excentricities of the orbits and their inclinations are 
very small. But the integrals (4), (5), (6), (7), which are given in No. 
471, give the same relations, whatever may be the excentricities and in- 

x d v ~ "~ v d x . 

clinations. For this, we shall observe that . * is double the 

d t 

area described during the instant d t, by the projection of the radius- 
vector of the planet fj> upon the plane of x, y. In the elliptic motion, if 
we neglect the mass of the planet as nothing compared with that of the 
sun, taken for unity, we shall have, by the Nos. 157, 237, relatively to the 
plane of p s orbit, 



. 

In order to refer the area upon the orbit to the fixed plane, we must 
multiply by the cosine of the inclination f of the orbit to this plane ; we 
shall, therefore, have, with reference to this plane, 

e s ) 



xdy ydx . ^ - ^ /( 

- J / - = cos. <p V a (1 e 2 ) = . / = ^ 
d t v <V 1 + 



tan. 



15-6 A COMMENTARY ON [SECT. XI. 

In like manner 

x dy y dx _ la (I e 2 ) . 
d t = V 1 + tan. 2 p 

&c. 

These values of x d y y d x, x d y y d x , &c. may be used, 
abstraction being made of the inequalities of the motion of the planets, 
provided we consider the elements e, e , &c. <p, <p f , Sec. as variables, in 
virtue of the secular inequalities; the equation (4) of No. 471 will there 
fore give in that case, 

a (I e 2 ) , , /a (l e /2 ) , 
C = ^Vl + tan. P + * */l + tan.V + &C 

j(x -.x)(d y -dy)-(y - y )d* -clx)l 
" *** \ d t J 

Neglecting this last term, which always remains of the order ^ p , we 
shall have 

a (l e 2 ) 



c = 



Thus, whatever may be the changes which the lapse of time produces 
in the values of e, e , &c. <p, <p , &c. by reason of the secular variations, 
these values ought always to satisfy the preceding equation. 

If we neglect the small quantities of the order e 4 , or e 2 p 5 , this equa 
tion will give 

c = (j, V a + A* V a + &c. | ft, V a {c 2 + tan. * p] 

A* V a fe 2 + tan 2 p 7 } &c. ; 

and consequently, if we neglect the squares of e, e , p, &c. we shall have 
P V a -\- (* V a + &c. constant. We have seen above, that if we only 
retain the first power of the perturbing force, a, a , &c. will be separately 
constant ; the preceding equation will therefore give, neglecting small 
quantities of the order e 4 or e 2 p 8 , 

const. = fj. V a {e 2 + tan. 2 <p] + /j, V a {e 2 + tan. 2 <p\ + &c. 

On the supposition that the orbits are nearly circular, and but little 
inclined to one another, the secular variations are determined (No. 522) 
by means of differential equations independent of the inclinations, and 
which consequently are the same as though the orbits were in one plane. 
But in this hypothesis we have 

p = 0, = 0, &c. 
the preceding equation thus becoming 

constant = e 2 /* V a + e 2 ^ V a + e" 2 p." V a" + &c. 
an equation already given in No. 524. 



BOOK L] NEWTON S PRINCIPIA. 15T 

In like manner the secular variations of the inclinations of the orbits, 
are (No. 526) determined by means of differential equations, independent 
of excentricities, and which consequently are the same as though the or 
bits were circular. But in this hypothesis we have e = 0, e 0, &c. 
Wherefore 

const.=/.i \/a . tan 2 >+ t* Va. . tan. 2 ? +,," Va" . tan. " <p" + &c. 
an equation which has already been given in No. 526. 

If we suppose, as in the last No. 

p = tan. <p sin. 6 ; q = tan. <p cos. 6 ; 

it is easy to prove that, the inclination of the orbit of & to the plane of 
x, y being (p, and the longitude of its ascending node reckoned from the 
axis of x being 0, the cosine of the inclination of this orbit to the plane of 
x, z, will be 

q 

V ( 1 + tan. 2 p) 

y ~"^ - 



Multiplying this quantity by - ~" - , or by its value Va.(l e 2 ), 

Cl L 

v f I TJ r ___ y /"J Y 

we shall have the value of -- -, - ; the equation (5) of No. 471, 

Cl L 

will therefore give us, neglecting quantities of the order & 2 , 

a (1 e 2 ) , , /a . (1 e /2 ) 



C = 



We shall find, in like manner, that the equation (6) of No. 471, gives 



If in these two equations we neglect quantities of the order e* or e s <f> ; 
they will become 

const. = i* q . V a + pf q V a + &c. 
const. = ft p V a + // p r V a + &c. 
equations already found in No. 526. 

Finally, the equation (7) of No. 471, will give, observing that by 478, 
m_ _ 2 m __ d x 8 + dy 2 + dz 2 
V g d t z 

and neglecting quantities of the order p (* , 

const. = + ^ + ^ + &c. 

These duTerent equations subsist, when we regard inequalities due to 
very long periods, which affect the elements of the orbits of ^ p , &c. 
We have observed in No. 521, that the relation of the mean motions of 
these bodies may introduce into the expressions of the axis-majors of the 



158 A COMMENTARY ON [SECT. XI. 

orbits considered variable, inequalities whose arguments proportional to 
the time increase very slowly, and which having for divisors the coeffi" 
cients of the time t, in these arguments, may become sensible. But it is 
evident that, retaining the terms only which have like divisors, and consi 
dering the orbits as ellipses whose elements vary by reason of those terms, 
the integrals (4), (5), (6), (7), of No. 471, will always give the relations 
between these elements already found; because the terms of the order 
/u, (if which have been neglected in these integrals, to obtain the relations, 
have not for divisors the very small coefficients above mentioned, or at 
least they contain them only when multiplied by a power of the perturb 
ing forces superior to that which we are considering. 

529. We have observed already, that in the motion of a system of 
bodies, there exists an invariable plane, or such as always is of a 
parallel situation, which it is easy to find at all times by this condition, that 
the sum of the masses of the system, multiplied respectively by the pro 
jections of the areas described by the radius-vectors in a given time is a 
maximum. It is principally in the theory of the solar system, that the re 
search of this plane is important, when viewed with reference to the proper 
motions of the stars and of the ecliptic, which make it so difficult to astro 
nomers to determine precisely the celestial motions. If we call 7 the 
inclination of this invariable plane to that of x, y, and n the longitude of 
its ascending node, it is easily found that 

c" 
tan. /sin. rirr ; tan. y cos. 

and consequently that 



u.Va(l e 2 ) sin. p sin. 0-fVvV (1 e /2 ) sin. p sin. 
tan.y sin. n z= - - * == - - - - - : - 

(\ e 2 )cos. p+^ Va (l e 2 ) cos. 



_, e 2 ). sin. pcos. 6-\-(jf V& (\ e /2 ) sin. p cos.0 +&c. 

" 7 * 



(1 e 2 ) .cos. f + ^V a (l e 2 ) .cos. 
We shall determine very easily, by means of these values, the angles 7 
and n. We see that to determine the invariable plane we ought to know 
the masses of the comets, and the elements of their orbits ; fortunately 
these masses appear to be so very small that we may, without sensible 
error, neglect their action upon the planets : but time alone can clear up 
this point to us. We may observe here, that relatively to this invariable 
plane the values of p, q, p , q , &c. contain no constant tei ms ; for it is 
evident by the equations (C) of No. 526, that these terms are the same for 
p, p , p", &c. and that they are also the same for q, q , q", &c. ; and since re 
latively to the invariable plane, the constants of the first members of the 



BOOK L] NEWTON S PRINCIPIA. 159 

equations (1) and (2) of No. 526 are nothing: the constant terms disap 
pear, by reason of these equations, from the expressions p, p , &c. 
q, q , &c. 

Let us consider the motion of the two orbits, supposing them inclined 
to one another, by any angle whatever : we shall have by No. 528, 

c sin. <p cos. 6 . p V a ( 1 e 2 ) + sin. <f> . cos. 6 . (jf V af (1 e 2 ) ; 

c" = sin. <p sin. 6 . i* V a (1 e 2 ) +sin. <p . sin. (f . ,</ V a (I e a ). 

Let us suppose that the fixed plane to which we refer the motion of the 
orbits, is the invariable plane of which we have spoken, and by reference 
to which the constants of the first members of these equations, are no 
thing, as may easily be shown. The angles <p and <p being positive, the 
preceding equations give the following ones : 

p V a (1 e 2 ) . sin. <p =//V~a (1 e /2 ) . sin. <f> ; 
sin. 6 = sin. 6 ; cos. = cos. ^ ; 

whence we derive 6 = 6 + the semi circumference ; the nodes of the or 
bits are consequently upon the same line ; but the ascending node of the 
one coincides with the decending node of the other ; so that the mutual 
inclination of the two orbits is equal to <p + <p . 

We have by No. 528, 



c = / V a ( 1 e 2 ). cos. <p + ft/ V a ( 1 e 2 ) cos. $ ; 
by combining this equation with the preceding one between sin. <p and 
sin. p , we shall have 



os.p. V a(l _e 2 )=c 2 +At 2 a(l e 2 ) i* *. a (l e 2 ). 
If we suppose the orbits circular, or at least having excentricity so small 
that we may neglect the squares of their excentricities, the preceding 
equation will give p constant : for the same reason <p f will be constant ; the 
inclinations of the planes of the orbits to the fixed plane, and to one ano 
ther, will therefore be constant, and these three planes will always have a 
common intersection. It thence results that the mean instantaneous va 
riation of this intersection is always the same ; because it can only be a 
function of these inclinations. When they are very small, we shall easily 
find by No. 528, and in virtue of the preceding relation between sin. <p 
and sin. p , that for the time t, the motion of this intersection is 

{(0,1) + (1,0)}. t. 

The position of the invariable plane to which we refer the motion of 
the orbits, may easily be determined for any instant whatever ; for we 
have only to divide the angle of the mutual inclination of the orbits into 
two angles, <p and <f> , such as that we have in the preceding equation be- 



130 A COMMENTARY ON [SECT. XI. 

tween sin. <p and sin. <p r . Designating, therefore, this mutual inclination 
by w, we shall have 

// V a! (1 e /2 ). sin. 
tan. p = 



a (1 e 2 ) + iff V a (1 e /2 ) . cos. 

SECOND METHOD OF APPROXIMATION OF THE CELESTIAL MOTIONS. 

530. We have already seen that the coordinates of the celestial bodies, 
referred to the foci of the principal forces which animate them, are deter 
mined by differential equations of the second order. We have integrated 
these equations in retaining only the principal forces, and we have shown 
that in this case, the orbits are conic sections whose elements are the 
arbitrary constants introduced by integration. 

The perturbing forces adding only small inequalities to the elliptic mo 
tion, it is natural to seek to reduce to the laws of this motion the troubled 
motion of the celestial bodies. If we apply to the differential equations 
of elliptic motion, augmented by the small terms due to the perturbing 
forces, the method exposed in No. 512, we can also consider the celestial 
motions in orbits which turn into themselves, as being elliptic; but the 
elements of this motion will be variable, and by this method we shall ob 
tain their variations. Hence it results that the equations of motion, being 
differentials of the second order, not only their finite integrals, but also 
their infinitely small integrals of the first order, are the same as in the 
case of invariable ellipses ; so that we may differentiate the finite equa 
tions of elliptic motion, in treating the elements of this motion as con 
stant. It also results from the same method that the differential equa 
tions of the first order may be differentiated, by making vary only the 
elements of the orbits, and the first differences of the coordinates ; pro 
vided that instead of the second differences of these coordinates, we sub 
stitute only that part of their values which is due to their perturbing 
forces. These results can be derived immediately from the consideration 
of elliptic motion. 

For that purpose, conceive an ellipse passing through a planet, and 
through the element of the curve which it describes, and whose focus is 
occupied by the sun. This ellipse is that which the planet would invari 
ably describe, if the perturbing forces were to cease to act upon it. Its 
elements are constant during the instant d t; but they vary from one 
instant to another. Let therefore V = 0, be a finite equation to an in 
variable ellipse, V being a function of the rectangular coordinates x, y, z 






BOOK I.] NEWTON S PRINCIPIA. 161 

and the parameters c, c , &c. which are functions of the elements of ellip 
tic motion. Since, however, this ellipse belongs to the element of the 
curve described by the planet during the instant d t ; the equation V = 
will still hold good for the first and last point of this element, by regard 
ing c, c , &c. as constant. We may, therefore, differentiate this equation 
once in only supposing x, y, z, to vary, which gives 

0= (, ^ d x + ( j ) d y + (-p ) d z; (i) 
\d x / \d y / ^d z / 

We also see the reason why the finite equations of the invariable el 
lipse, may, in the case of the variable ellipse, be differentiated once in 
treating the parameters as constant. For the same reason, every differ 
ential equation of the first order relative to the invariable ellipse, equally 
holds good for the variable ellipse ; for let V = be an equation of this 

order, V being a function of x, y, z, -T , -s-4- , -T , and the parameters 

c, c , &c. It is clear that all these quantities are the same for the varia 
ble ellipse as well as for the invariable ellipse, which for the instant d t 
coincides with it. 

Now if we consider the planet, at the end of the instant d t, or at the 
commencement of the following one ; the function V will vary from the 
ellipse relative to the instant d t to the consecutive ellipse only by the 
variation of the parameters, since the coordinates x, y, z, relative to the 
end of the first instant are the same for the two ellipses ; thus the function 
V being nothing, we have 



This equation may be deduced from the equation V = 0, by making 
x, y, z, c, c , &c. vary together ; for if we take the differential equation 
(i) from this differential, we shall have the equation (i ). 

Differentiating the equation (i), we shall have a new equation in d c, 
d c , &c. which with the equation (i ) will serve to determine the parame 
ters c, c , &c. Thus it is that the geometers, who were first occupied in 
the theory of celestial perturbations, have determined the variations of 
the nodes and the inclinations of the orbits : but we may simplify this 
differentiation in the following manner. 

Consider generally the differential equation of the first order V 7 = 0, 
an equation which belongs equally to the variable ellipse, and to the in 
variable ellipse which, in the instant d t, coincides with it. In the follow 
ing instant, this equation belongs also to the two ellipses, but with this 

Vor.. II. L 



162 A COMMENTARY ON [SECT. XI. 

difference, that c, c , &c. remain the same in the case of the invariable 
ellipse, but vary with the variable ellipse. Let .V be what V becomes, 
when the ellipse is supposed invariable, and V/ what this same function 
becomes in the case of the variable ellipse. It is clear that in order to 
have V we must change in V, the coordinates x, y, z, which are rela 
tive to the commencement of the first instant d t, in those which are rela 
tive to the commencement of the second instant; we must then augment 
the first differences d x, d y, d z respectively by the quantities d 2 x, d 2 y 
d 2 z, relative to the invariable ellipse, the element d t of the time, being 
supposed constant. 

In like manner, to get V/, we must change in V, the coordinates 
x, y, z, in those which are relative to the commencement of the second 
instant, and which are also the same in the two ellipses ; we must then 
augment d x, d y, d z respectively by the quantities d 2 x, d 2 y, d 2 z ; finally, 
we must change the parameters c, c , &c. into c + d c, c + d c ; &c. 

The values of d 2 x, d 2 y, d 2 z are not the same in the two ellipses ; 
they are augmented, in the case of the variable ellipse, by the quantities 
due to the perturbing forces. We see also that the two functions V" 
and V/j differing only in this that in the second the parameters c, c , &c. 
increase by d c, d c , &c. ; and the values of d 2 x, d 2 y, d 2 z relative to 
the invariable ellipse, are augmented by quantities due to the perturbing 
forces. We shall, therefore, form V/ V", by differentiating V in the 
supposition that x, y, z are constant, and that d x, d y, d z, c, c , &c. 
are variable, provided that in this differential we substitute for d 2 x, d 2 y, 
d 2 z, &c. the parts of their values due solely to the disturbing forces. 

If, however, in the function V" V we substitute for d 2 x, d z y, d 2 z 
their values relative to elliptic motion, we shall have a function of x, y, z, 

-: , -j-^- , -: , c, c , &c., which in the case of the invariable ellipse, is 
d t d t d t 

nothing; this function is therefore also nothing in the case of the variable 
ellipse. We evidently have in this last case, V/ V = 0, since this 
equation is the differential of the equation V = : taking it from the 
equation V/ V = 0, we have V/ V" = 0. Thus, we may, in this 
case, differentiate the equation V = 0, supposing d x, d y, d z, c, c , &c. 
alone to vary, provided that we substitute for d 2 x, d - y, d 2 z, the parts 
of their values relative to the disturbing forces. These results are exactly 
the same as those which we obtained in No. 512, by considerations purely 
analytical ; but as is due to their importance, we shall here again present 
them, deduced from the consideration of elliptic motion. 



BOOK L] 



NEWTON S PRINCIPIA. 



163 



531. Let us resume the equations (P) of No. 513, 

-*!15 J. * 4 
U ~ + 



-JT 2 " -jr 
d 2 z m z 

= dT 2 ; ~p~ 

If we suppose R = 0, we shall have the equations of elliptic motion, 
which we have integrated in (478)- We have there obtained the seven 
following integrals 
xdy vdx 



c ~ 


dt 
x d z z d 


3 

X 


f y d y . d x 


z d z .d x -* 


c" 


dt 
y d z z d 


y. 


= 
= 

- 


dt 
f 4- x! m - 


dy 2 + dz 2 \ 


1 T * | 

PI f m 


dt 2 I 

da . 1 n \ 
x z + d z z ^ 


d t 2 
x d x . d y 


dt 2 
z d z . d y 


f J.\- - 


dt 2 / 

dx 2 -j-dy z ^ 


dt 2 
x d x .d z 


dt 2 
L y d y . d z i 


m 2m 


dt 2 J 
d x 2 + d y* + 


d z 2 


dt 2 



(P) 



These integrals give the arbitrages in functions of their first differences; 
they are under a very commodious form for determining the variations of 
these arbitraries. The three first integrals give, by differentiatino- them, 
and making vary by the preceding No. the parameters c, c/, c", and the 
first differences of the coordinates, 

, x d 2 y y d 2 x 

d c = J , -? 

d t 



dc = 



x d 2 z z d 2 x 
" dT~ 



, y d 2 z z d 2 v 

d c"= * = ^~ 

dt 



Substituting for d 2 x, d 2 y, d 2 z, the parts of their values due to the 
perturbing forces, and which by the differential equations (P) are 



164 A COMMENTARY ON [SECT. XI. 

we shall have 



, dR 

dc = 



d R\ /a R% 



We know from 478, 479 that the parameters c, c , c" determine three 
elements of the elliptic orbit, viz., the inclination <p of the orbit to the 
plane of x, y, and the longitude 6 of its ascending node, by means of the 
equations 

V (c 2 + c" 2 ) c" - 

tan. <p = s - 21 ; tan. 6 = , ; 

and the semi-parameter a (1 e 2 ) of the ellipse by means of the equa 
tion 

ma (l e 2 ) = c 2 +c /2 + c" J . 

The same equations subsist also in the case of the variable ellipse, 
provided we determine c, c , c" by means of the preceding differential 
equations. We shall thus have the parameter of the variable ellipse, its 
inclination to the fixed plane of x, y and the position of its node. 

The three first of the equations (p) have given us in No. (479) the 

finite integral 

= c" x c y + c z : 
this equation subsists in the case of the troubled ellipse, as also its first 

difference 

= c" d x c a y + c d z 

taken in considering c, c , c" constant. 

If we differentiate the fourth, the fifth and the sixth of the integrals 
(p), making only the parameters f, f , f", and the differences d x, d y, d z 
vary; if moreover, we substitute then for d 2 x, d 2 y, d 2 z, the quantities 

" R \ d 1 2 (ilh d t 2 (\ we shall have 
V>~~ at \d v) 9 Viz;" 



+ (x d y - y d x) () + (z d y - y d z) , 



BOOK I.] NEWTON S PRINCIPIA. 165 

Rxl , f /dR 



df = d 

+ (X d Z Z d X) (g^) + (y d Z Z d y) (gy). 

Finally, the seventh of the integrals (p) ?1 differentiated in the same 
manner, will give the variation of the semi-axis-major a, by means of the 
equation 

d. ~ = 2dR, 

the differential d R being taken relatively to the coordinates x, y, z, alone 
of the body /*. 

The values of f, P, f" determine the longitude of the projection of the 
perihelion of the orbit, upon the fixed plane, and the relation of the ex- 
centricity to the semi-axis-major ; for I being the longitude of this projec 
tion by (479) we have 

p 

tan. I = ->-; 

and e being the ratio of the excentricity to the semi -axis-major, we have 

me = V (f 2 + f 2 + f" 2 )- 

This ratio may also be determined by dividing the semi-parameter 
a (1 e 2 ), by the semi-axis-major a : the quotient taken from unity will 
give the value of e z . 

The integrals (p) have given by elimination (479) the finite integral 

= m g h 2 + f x + f .y + f" z : 

this equation subsists in the case of the troubled ellipse, and it determines 
at each instant, the nature of the variable ellipse. We may differentiate 
it, considering f, f , f" as constant ; which gives 

= m d s + f d x + f d y + f" d z. 

The semi-axis-major a gives the mean motion of /A, or more exactly, 
that which in the troubled orbit, corresponds to the mean motion in the 

invariable orbit ; for we have (479) n = a ~ 2 V m ; moreover, if we de 
note by < the mean motion of /t*, we have in the invariable elliptic orbit 
d = n d t : this equation equally holds good in the variable ellipse, 
since it is a differential of the first order. Differentiating we shall have 
d * = d n . d t ; but we have 

San ,m 3anrfR 
d n = -- . d . - = -- , 
2m a m 

therefore 

3 a n d t. d R 



d * I = 

m 



L3 



J66 A COMMENTARY ON [SECT. XT. 

and integrating 

= - .//a n d t . d R. 
m JJ 

Finally we have seen in (No. 473) that the integrals (p) are equivalent 
to but five distinct integrals, and that they give between the seven para 
meters c, c , c", f, f, i" e, the two equations of condition 
= f c " f c + PC; 



: 



in f* + f /2 + f" 2 m 



a c 2 + c 2 + c //2 

these equations subsist therefore in the case of the variable ellipse provid 
ed that the parameters are determined as above. 

We can easily verify these statements a posteriori. 

We have determined five elements of the variable orbit, viz., its inclin 
ation, position of the nodes, its semi-axis-major which gives its mean mo 
tion, its excentricity and the position of the perihelion. It remains for us 
to find the sixth element of elliptic motion, that which in the invariable 
ellipse corresponds to the position of 11 at a given epoch. For this pur 
pose let us resume the expression of d t (473) 

dt Vm _ d v(l e 2 )*" 
a f = {1 + ecos. (v )} 

This equation developed into series gives (473) 
n d t = d v {] + E (1 > cos. (v ) + E cos. 2 (v ) + &c.J, 

Integrating this equation on the supposition of e and w being con 
stant, we shall have 

E C1) 
/n d t + e - v + E C1 ) sin. (v ) -f -5- sin. 2. (v ) + &c. 

tQ 

being an arbitrary. This integral is relative to the invariable ellipse : 
to extend it to the variable ellipse, in making every thing vary even to 
the arbitrages, E, e, & which it contains, its differential must coincide with 
the preceding one ; which gives 

da = de{ (^ e -)sin. (v w; + * (-^) sin. 2 (v - w) + &c.} 

d fcEWcos. (v .) + Ecos.2(v ) + &c.} 
v ro being the true anomaly of (A measured upon the orbit, and the 
longitude of the perihelion also measured upon the orbit, We have de 
termined above, the longitude I of the projection of the perihelion upon 
a fixed plane. But by (488) we have, in changing v into -a and v, into I 
in the expression of v [3 of this No. 

* 8 = I 6 + tan. * $ <p sin. 2(1 6} + &c. 



BOOK I.] NEWTON S PRINCIPIA. 167 

Supposing next that v, v /5 are zero in this same expression, we have 
|8 = + tan. 2 <f> sin. 2 6 + &c. 

wherefore, 

*r = I + tan. 8 | p. {sin. 2 + sin. 2 (I 6) + &c.} 

which gives 

d = dl. {1 + 2 tan. 2 |f cos. 2 (I 6) + &c.J 

+ 2 d tan. 2 p {cos. 2 d cos. 2 (I 6} + bcc.} 

dp tan. $p ^ {s[ ^ 2 , sin> 2 (I _ 0) + &c . } . 

cos. ^ p 

Thus the values of d I, d 0, and d p being determined by the above, we 
shall have that of d v ; whence we shall obtain the value of d . 

It follows from thence that the expressions in series, of the radius-vec 
tor, of its projection upon the fixed plane, of the longitude whether re 
ferred to the fixed plane or to the orbit, and of the latitude which we 
have given in (No. 488) for the case of the invariable ellipse, subsist equal 
ly in the case of the troubled ellipse, provided we change n t into/n d t, 
and we determine the elements of the variable ellipse by the preceding 
formulas. For since the finite equations between g, v, s, x, y, z, and 
J n d t, are the same in the two cases, and because the series of No. 488 
result from these equations, by analytical operations entirely independent 
of the constancy or variability of the elements, it is evident these expres 
sions subsist in the case of variable elements. 

When the ellipses are very excentric, as is the case with the orbits of 
the comets, we must make a slight change in the preceding analysis. The 
inclination <p of the orbit to the fixed plane, the longitude 6 of its ascend 
ing node, the semi-axis-major a, the semi-parameter a (1 e 2 ), the ex- 
centricity e, and the longitude I of the perihelion upon the fixed plane 
may be determined by what precedes. But the values of -a and of d -a 
being given in series ordered according to the powers of tan. \ p, in order 
to render them convergent, we must choose the fixed plane, so as to make 
tan. \ p inconsiderable ; and to effect this most simply is to take, for the 
fixed plane, that of the orbit of ^ at a given epoch. 

The preceding value of d E is expressed by a series which is convergent 
only in the case where the excentricity of the orbit is inconsiderable, we 
cannot therefore make use of it in this case. Instead, let us resume the 
equation 

d t V m d v ( 1 e 2 ) $ 

~f~ = [I + ecos. (v ~)} 2 



168 A COMMENTARY ON [SECT. XL 

If we make 1 e = a, \ve have by (489) in the case of the invariable 
ellipse, 



T being an arbitrary. To extend this equation to the variable ellipse, 
we must differentiate it by making vary T, the semi parameter a ( 1 e 2 ), 
, and v. We shall thence obtain a differential equation which will de 
termine T, and the finite equations which subsist in the case of the in 
variable ellipse, will still hold good in that of the variable ellipse. 

532. Let us consider more particularly the variations of the elements 
of ft s orbit, in the case of the orbits being of small excentricity and but 
little inclined to one another. We have given in No. 515. the manner of 
developing R in a series of sines and cosines of the form 

(jf k cos. (i n t i n t + A) 

k and A being functions of the excentricity and inclinations of the orbits, 
the positions of their nodes and perihelions, the longitudes of the bodies 
at a given epoch, and the major-axes. When the ellipses are variable 
all these quantities must be supposed to vary conformably to what pre 
cedes. We must moreover change in the preceding term, the angle 
i n t i n t into \ J n d t i J n d t, or which is tantamount, into 
i % - i . 

However, by the preceding No., we have 



The difference d R being taken relatively to the coordinates x, y, z, 
of the body p, we must only make vary, in the term 

(t! k cos. (i i C + A) 

of the expression of R developed into a series, what depends upon the 
motion of this body ; moreover, R being a finite function of x, y, z, x , y , z 
we may by No. 530, suppose the elements of the orbit constant in the 
difference d R ; it suffices therefore to make vary in the preceding term, 
and since the difference of is n d t, we have 

i (if. k n d t . sin. (V % i + A) 

for the term of d R which corresponds to the preceding term of R. Thus, 
with respect to this term only, we have 

! " 2l/// / k nd t.sin. (i i + A); 



m 



BOOK I.] NEWTON S PRINCIPIA. 169 

- ^-ffa k n 2 d t 2 sin. (i i + A). 

If we neglect the squares and products of the perturbing masses, we 
may, in the integrals of the above terms, suppose the elements of elliptic 
motion constant. Hence becomes n t and , n t ; whence we get 
1 2 i y! n k 



;c- - \ 
m (i n i n) 



.., . A N 

cos - (i n t i n t + A) 



3 i // a n 2 k . ,./ / . \\ 

I = -- r^-, - r-^s sin. (i n t i n t + A). 
m (i n in) 2 

Hence we perceive that if i n in is not zero, the quantities a and 
only contain periodic inequalities, retaining only the first power of the 
perturbing force ; but i and i being whole numbers, the equation i n in 
= cannot subsist when the mean motions of p and (t! are incommen 
surable, which is the case with the planets, and which can be admitted 
generally, since n and n being arbitrary constants susceptible of all possi 
ble values, their exact relation of number to number is not at all probable. 

We are, therefore, conducted to this remarkable result, viz., that the 
principal axes of the planets, and their mean motions, are only subject to 
periodic inequalities depending on their configuration, and that thus in ne 
glecting these quantities, their principal axes are constant and their mean 
motions uniform, a result agreeing "with what has otherwise been found by 
No. 521. 

If the mean motions n t and n t, without being exactly commensurable, 
approach very nearly to the ratio i : i ; the divisor i n in is very" 
small, and there may result in and inequalities, which increasing very 
slowly, may give reason for observers to suppose that the mean motions 
of the two bodies p, (i! are not uniform. We shall see, in the theory of 
Jupiter and Saturn, that this is actually the case with regard to these two 
planets : their mean motions are such that twice that of Jupiter is very nearly 
equal to five times that of Saturn ; so that 5 n 2 n is hardly the sixty- 
fourth part of n. The smallness of this divisor, renders very sensible the 
term of the expression for , depending upon the angle 5 n t 2 n t, 
although it is of the order i i, or of the third order, relatively to the 
excentricities and inclinations of the orbits, as we have seen in No. 515. 
The preceding analysis gives the most sensible part of these inequalities ; 
for the variation of the mean longitude depends on two integrations, whilst 
the variations of the other elements of elliptic motion depend only on 
one integration ; only terms of the expression of the mean longitude can 
therefore have the divisor (i n in) 2 ; consequently with regard only 



A COMMENTARY ON [SECT. XL 

to these terms, which, considering the smallness of the divisor ought to 
be the more considerable, it will suffice, in the expressions of the radius- 
vector, the longitude and latitude, to derive from these terms, the mean 
longitude. 

When we have inequalities of this kind, which the action of f produces 
in the mean motion of /*, it is easy thence to get the corresponding ine 
qualities which the action of p produces in the mean motion of /* In 
fact, if we have regard only to the mutual action of three bodies M, ^ and 
/* ; the formula (7) of (471) gives 

const = ,dx- + dy + d Z dx" + dy"+d* 

ilt 2 - dt 2 

_ (ft, d x + ft d x ) 2 + (ft, d y + p d y ) 2 + (0 d z + ft, d z ) 2 
(M +|t6 + p )*d t 2 



2 My 2 M 






+ z* V(x -x) 2 -f(y y)M-(z -z) 
The last of the integrals (p) of the preceding No. gives, by substituting 

for the integral 2fd R, 

dx 2 -f-dy 2 + dz 2 _ 2 (M -f ^) 



If we then call R , what R becomes when we consider the action of 
upon |tt , we shall have 

R , _ y. (x x r + y y ; + z zQ _ p 

(x+y + z)* V"(? x) 2 +(y -y) 2 +(z z)^ 

dz^_ 2 (M + ft ) 

" 



dt 2 

the differential characteristic ^ only belonging to the coordinates of the 

i i / cur*.- f dx 2 + dy 2 + dz 2 , d x /2 + d y /2 + d z /2 
body /* . Substituting for -- - -*| - - and - , J - -- 

U. L Cl t 

the values in the equation (a), we shall have 



- const 
_ const. 



2 (M + ,* + /* ) dt 2 

2 / 2 

+ " , z - 2 g H - /2 /8 ^ . 

It is evident that the second member of this equation contains no terms 
oi the order of squares and products of the ^ & , which have the divisor 
i n in; relative, therefore, only to these terms, we shall have 

I -j- f^ J d R = 0; 



BOOK I.] NEWTON S PRINCIPIA. 171 

thus, by only considering the terms which have the divisor (V n in) 2 , 
we shall have 

3/yVn dt.d R _ _ p(M + v).afjS Sffa n d t . d R 

M + (* ~ iif (M + /TTn ~ M. + P 

but we have 

Sffandt.dR , _ Bffaf n d t . d R 
^ = ~~M + p ; ^ = M + ^ 

we therefore get 

^ (M + ^) a n % + p (M + /*) a n 7 = 0. 
Again, we have 

_ V (M + AQ . _V (M + ^0. 

a * a 2 

neglecting therefore /A, /" , in comparison with M, we shall have 

A* V a . + fit V a . = ; 
or 



v s^ r 

Thus the inequalities of , which have the divisor (i n 7 in) 2 , give 
us those of , which have the same divisor. These inequalities are, as 
we see, affected with the contrary sign, if n and n have the same sign, or 
which amounts to the same, if the two bodies /* and (i! circulate in the 
same direction; they are, moreover, in a constant ratio; whence it follows 
that if they seem to accelerate the mean motion of /u-, they appear to re 
tard that of (*> according to the same law, and the apparent acceleration 
of jw, will be to the apparent retardation of /", , as p f V af is to / V a. The 
acceleration of the mean motion of Jupiter and the retardation of that of 
Saturn, which the comparison of modern with ancient observations made 
known to Halley, being very nearly in this ratio ; it may be concluded 
from the preceding theorem, that they are due to the mutual action of the 
two planets; and, since it is constant, that this action cannot produce in 
the mean motions any alteration independent of the configuration of the 
planets, it is very probable that there exists in the theory of Jupiter and 
Saturn a great periodic inequality, of a very long period. Next, consider 
ing that five times the mean motion of Saturn, minus twice that of Jupi 
ter is very nearly equal to nothing, it seems very probable that the phe 
nomenon observed by Halley, was due to an inequality depending upon 
this argument. The determination of this inequality will verify the con 
jecture. 

The period of the argument i n t i n t being supposed very long, 



172 A COMMENTARY ON [SECT. XI. 

the elements of the orbits of /*, and /// undergo, in this interval sensible 
variations, which must be taken into account in the double integral 
ffa k n 2 d t 2 sin. (V n t i n t + A). 

For that purpose we shall give to the function k sin. (i n t i n t + A), 
the form 

Q sin. (i n t i n t + i e i ) + Q cos. (i n t i n t + i i if) 
Q and Q being functions of the elements of the orbits : thus we shall 
have 

ffa. k n 2 d t 2 sin. (i n 1 i n t + A) = 

n 2 a sin. (V n t i n t + iV-i / o 2 d Q 3d 2 Q 1 

(V n i n) 2 * X (iV in)dt (i n in) s dt "*" C * J 

n 2 a cos.(iVt i n t+i t i Q f o , 2 d Q 3 d 2 Q 1 

(I 7 n i np * t W (i n in)dt (i n in) 2 dt + C ) 

The terms of these two series decreasing very rapidly, with regard to 
the slowness of the secular variations of the elliptic elements, we may, in 
each series, stop at the two first terms. Then substituting for the ele 
ments of the orbits their values ordered according to the powers of the 
tune, and only retaining the first power, the double integral above may 
be transformed in one term to the form 

(F + E t) sin. (i n t i n t + A + H t). 

Relatively to Jupiter and Saturn, this expression may serve for many 
ages before and after the instant from which we date the given epoch. 

The great inequalities above referred to, become sensible amongst the 
terms depending upon the second power of the perturbing forces. In 
fact, if in the formula 

= ^~ff^ k n 2 . d t 2 . sin. <i % i + A), 

we substitute for , g their values 

3 i & a n 2 k . ,.,/.. 

n t 777-7 r-r- z sin. (i n t i n t + A) ; 

m(i / n / in) z 

3 i / a n 2 k /a . ,-, , . ^ 

n t 7TJ-. = ./ - sin. (i n t i n t + A), 

1x1(1 n in)W a 

there will result among the terms of the order (j, z , the following 

9iV 2 a 2 n 4 k 2 i // V a + r> V a . . 

pi 0,., , = a 7-h sm - * (i n t i n t + A). 

8 m 8 (Y n 7 i n) 4 ^ V a 

The value of % contains the corresponding term, which is to the one 
preceding in the ratio v> V a : (if V a , viz. 

9iV 2 a 2 n 4 k 2 ,. ,/,,-, / , ft V a . Q/w ,. . A , 

8m 2 (i n -in) 4 ^ V a + i> ^ } ^F^ in. 2 (i n t-i n t + A). 

533. It may happen that the inequalities of the mean motion which are the 



BOOK I.] NEW T TON S PRINCIPIA. 173 

most sensible, are only to be found among terms of the order of the 
squares of the perturbing masses. If we consider three bodies, /*, AS /*" 
circulating around M, the expression of d R relative to terms of this or 
der, will contain inequalities of the form 

k sin. (i n t i n t + I" n" t + A) 

but if we suppose the mean motions n t, n t, n" t such that in i n 
-f- \" n" is an extremely small fraction of n, there will result a very sensible 
inequality in the value of . This inequality may render rigorously equal 
to zero, the quantity in i n + i" n", and thus establish an equation of 
condition between the mean motions and the mean longitudes of the three 
bodies /-, ,/, y! . This very singular case exists in the system of Jupiter s 
satellites. We will give the analysis of it. 

If we take M for the mass-unit, and neglect ^ /* , &" in comparison with 
it, we shall have 

2 _ 1 1 1 

= a 3 = a 7 " 3 "a 77 " 3 

we have then 

d = n d t ; d = n d t ; d " = n" d t ; 
wherefore 



d 2 


3 ida 

_rr, Y\ 3 " 


dt 


2 a 2 


d* 


3 ^ da 


dt 


2 n a /2 


d 2 i" 


n"$ a 

2 a" 2 


dt 



We have seen in No. 528, that if we neglect the squares of the excen- 
tricities and inclinations of the orbits, we have 

const. = ii V a + (* . V a + y! V a" ; 
which gives 

= p + j!- a/ + " d a// 
V a V a V a" 

From these several equations, it is easy to get 
d 2 _3 $ (U 
d t 2 n a ~ 

d 2 T 3 p. n % n n"da 



d t 2 (j!. n n - n" a 8 

i " , _3 m.n"* n n da 
d t 2 * /*". n n n" a 2 



174 A COMMENTARY ON 

Finally the equation 



[SECT. XL 



R 



of No. 531, gives 



We have therefore only to determine d R. 

By No. 513, neglecting the squares and products of the inclinations of 
the orbits, we have 

R = ^-L cos. (v v) ^ ( 2 2 s / cos. (v v) + g 2 )~ * 



cos. (v" v) 



2 s f> cos. (v" v) 



If we develope this function in a series ordered according to the cosines 
of v v, v" v and their multiples ; we shall have an expression of 
this form 



COS. (V V) -f- (* (ft 



cos. 2 (v - v) 



- (0) 



(g, n (0) + ^"(ft f O (1) cos. (v" - v) + p," (g, / ) W cos. 2 (v" - v) 



ft f") (3) COS. 3 (V 7 V) + &C. ; 



whence we derive 



^ I 



cos. 2 (v v) + &c. 



LCOS. 2 (v" v) + &c. 

, / A* (ft f ) (1) sin. (V v) + 2 (f, ) W sin. 2(v v) + &c. 1 

v \ + ^/( f , / ) d)sin.(v" v) + 2,u"(ft f x/ > ^sin.2(v // _v) +&C. J . 
Suppose, conformably to what observations indicate in the system of 
the three first satellites of Jupiter, that n 2 n and n 2 n" are 
very small fractions of n, and that their difference n 2 n (n 2 n y ) 
or n 3 n + 2 n" is incomparably smaller than each of them. 

* 

It results from the expressions of - , and of d v of No. 517, that the 

action of// produces in the radius-vector and in the longitude of//, a very 
sensible inequality depending on the argument 2 (n t n t + * e). 
The terms relative to this inequality have the divisor 4 (n n) 2 n 2 , 



BOOK I.] NEWTON S PRINCIPIA. 175 

or (n 2 n ) (3 n 2 n ), and this divisor is very small, because of the 
smallness of the factor n 2 n . We also perceive, by the consideration 
of the same expressions, that the action of ^ produces in the radius- 
vector, and in the longitude of //, an inequality depending on the argu 
ment (n t n t + s E), and which having the divisor (n n) 2 n 2 , 
or n (n 2 n ), is very sensible. We see, in like manner, that the action 
of &" upon f! produces in the same quantities a considerable inequality 
depending upon the argument 2 (n" t n t + *" * ) Finally, we 
perceive that the action of yJ produces in the radius-vector and in the 
longitude of &" a considerable inequality depending upon the argument 
n" t n t + t" g. These inequalities were first recognised by obser 
vations ; we shall develope them at length in the Theory of Jupiter s Sa 
tellites. In the present question we may neglect them, relatively to other 
inequalities. We shall suppose, therefore, 

d g = [i! E cos. 2 (n t n t + ? 2) ; 
a v = 11! F sin. 2 (n t n t + e) ; 

If ft" E" cos. 2(n" t n t + s" t )+ft G cos. (n t nt + s - s) 
a v = (*" F" sin. 2(n"t ri t + i )+/" H sin. (n t n t + e) 
d z" = p G cos. (n" t n t + i" * ) 
d v" = it," H sin. (n" t n t + i" e). 

We must, however, substitute in the preceding expression of d R for 
fj v > g> v/ *" v// > the values of a 5 g, n t + s + 5 v, a + d g , n t+ + 5 V, 
a" + 3 / , n" t + s// + 3 V", and retain only the terms which depend upon 
the argument n t 3 n r t + 2 n" t + 3 t + 2 s". But it is easy to see 
that the substitution of the values of 8 ^ d v, 8 g", 3 v" cannot produce any 
such term. This is not the case with the substitution of the values of 
8 and 5 v : the term (i! (g, g ) W d v sin. (v 7 v) of the expression of 
d R, produces the following, 



, 



sin. (n t 3 n t + 2 n" t + t 3 tf + 2 *"). 
This is the only expression of the kind which the expression of d R 

<\ 

contains. The expressions of , and of 3 v of No. 517, applied to the 

action of // upon & , give, retaining only the terms which have the divisor 
n 2 n", and observing that n" is very nearly equal to ^ n , 



(n 2 n") (3 n 2 n") 



176 A COMMENTARY ON [SECT. XI- 

__ 2E" 

a 

we therefore have 

d n = ; n d * E-. | 2 (a r " - ( d ( t ? ") I 

2 t \ d a / ) 

Xsin. (n t 3n t + 2n // t + s3 g + 2i // )= I .^-?. 

a 4 

Substituting this value of - in the values of r-^ , -p-^- . , , and 

a 2 d t d t d t 

making for brevity s sake 



we shall have, since n is very nearly equal to 2 n , and n to 2 n", 
^ 2 3.511 + 2.ilL = / 3 n sin.(nt 3n t + 2n" t + e ~3 f 
or more exactly 



so that if we suppose 

V = ^ 3 ^ + 2 C + - 3 . + 2 s", 
we shall have 



The mean distances n, a , a", varying but little as also the quantity n, 
we may in this equation consider /3 n 2 , as a constant quantity. Integrat 
ing, we have 

-M- dV 

V c 2 |8 n 2 cos. V 

c being an arbitrary constant. The different values of which this con 
stant is susceptible, give rise to the three following cases. 

If c is positive and greater than + 2 (3 n 2 , the angle V will increase 
continually, and this ought to take place, if at the origin of the motion, 
(n 3 n + 2 n") 2 is greater than + 2 /3 n 2 (1 + cos. V), the upper or 
lower signs being taken according as (3 is positive or negative. It is easy 
to assure ourselves of this, and we shall see particularly in the theory of 
the satellites of Jupiter, that /3 is a positive quantity relatively to the three 
first satellites. Supposing therefore + v = or V, T being the semi cir 
cumference, we shall have 

d ~ 

V c + 2 n 2 cos. 



BOOK I.] NEWTON S PRINCIPIA. 177 



In the interval from -a to -a =r , the radical V c + 2 /3 n 2 cos. 



is greater than V 2 fi n 2 , when c is equal or greater than 2 /3 n 2 ; we 
have therefore in this interval -a > n t V 2 (3. Thus, the time t which the 

i T 

angle w employs in arriving from zero to a right angle is less than -- . 

/w H r > 

The value of /3 depends upon the masses, w, /<* , /M," ; the inequalities ob 
served in the three first satellites of Jupiter, and of which we spoke above, 
give, between their masses and that of Jupiter, relations from whence it 

results that -== i s under two years, as we shall see in the theory 

of these satellites ; thus the angle would employ less than two years to 
increase from zero to a right angle ; but the observations made upon Ju 
piter s satellites, give since their discovery, -a constantly nothing or insen 
sible; the case which we are examining is not therefore that of the three 
first satellites of Jupiter. 

If the constant c is less than + 2 /3 n 2 , the angle V will not oscillate ; 
it will never reach two right angles, if |8 is negative, because then the 
radical V c 2 j3 n z cos. V, becomes imaginary ; it will never be no 
thing if J3 is positive. In the first case its value will be alternately greater 
and less than zero ; in the second case it will be alternately greater and 
less than two right angles. All observations of the three first satellites of 
Jupiter, prove to us that this second case belongs to these stars ; thus the 
value of /3 ought to be positive relatively to them ; and since the theory 
of gravitation gives /3 positive, we may regard the phenomenon as a new 
confirmation of that theory. 

Let us resume the equation 

dl = - d " - 

V c + 2 13 n 2 cos. w 

The angle w being always very small, according to the observations, 
we may suppose cos. -a = 1 &* the preceding equation will give by 
integration 

tsr =: X sin. (n t V /3 + y) 

X and y being two arbitrary constants which observation alone can deter 
mine. Hitherto, it has not been recognised, a circumstance which proves 
it to be very small. 

From the preceding analysis result the following consequences. Since 
the angle n t + 3 n t + 2 n" t + s 3 + It oscillates being some 
times less and sometimes greater than two right angles, its mean value is 
VOL. II. M 



178 A COMMENTARY ON [SECT. XI. 

equal to two right angles ; we shall therefore have, regarding only mean 
quantities 

n 3 n + 2 n" = 

that is to say, that die mean motion of the Jlrst satellite, minus three times 
that of the second, plus twice that of the third, is exactly and constantly 
equal to zero. It is not necessary that this equality should subsist exactly 
at the origin, which would not in the least be probable ; it is sufficient 
that it did very nearly so, and that n 3 n + 2 n" has been less, ab 
straction being made of the sign, than X n V j8 : and then that the mutual 
attraction has rendered the equality rigorous. 

We have next t 3 s -f 2 i" equal to two right angles ; thus the mean 
longitude of the first satellite, minus three times that of the second, pins twice 
that of the third, is exactly and constantly equal to two right angles. 

From this theorem, the preceding values of < /, and of 8 v are reduci 
ble to the two following 

8 g = (p G f" E") cos. (n t n t + 
a v = (i* H P." F") sin. (n t n t + ) 

The two inequalities of the motion of (i! due to the actions of fi and of 
it* , merge consequently into one, and constantly remain so. 

It also results from this theorem, that the three first satellites can never 
be eclipsed at the same time. They cannot be seen together from Jupi 
ter neither in opposition nor in conjunction with the sun ; for the preced 
ing theorems subsist equally relative to the synodic mean motions, and to 
the synodic mean longitudes of the three satellites, as we may easily 
satisfy ourselves. These two theorems subsist, notwithstanding the alter 
ations which the mean motions of the satellites undergo, whether they 
arise from a cause similar to that which alters the mean motion of the 
moon, or whether from the resistance of a very rare medium. It is evi 
dent that these several causes only require that there should be added to 

the value of -* r , a quantity of the form of -rrT , and which shall only 
d t (it" 

become sensible by integrations ; supposing therefore V = it -a, and -a 
very small, the differential equation in V will become 



The period of the angle n t V jS being a very small number of years, 

f\ 2 1 

whilst the quantities contained in -p? are, either constant, or embrace 
many ages; by integrating the above equation we shall have 



BOOK I.] NEWTON S PRINCIPIA. 179 

6 2 4/ 
= X sin. (n t V /3 + 7 ) gn dt * 

Thus the value of will always be very small, and the secular equa 
tions of the mean motions of the three first satellites will always be order 
ed by the mutual action of these stars, so, that the secular equation of the 
first, plus twice that of the third, may be equal to three times that of the 
second. 

The preceding theorems give between the six constants n, n , n", 
s, e , t" two equations of condition which reduce these arbitraries to four ; 
but the two arbitraries X and y of the value of or replace them. This 
value is distributed among the three satellites, so, that calling p, p , p" the 
coefficients of sin. (n t V /3 + 7) in the expressions of v, v , v", these 

d 2 t d 2 T d 2 ?" 
coefficients are as the preceding values of -7 if 5 JTY 5 ~A~I? > an< * more 

over we have p 3 p + 2 p" = X. Hence results, in the mean mo 
tions of the three first satellites of Jupiter, an inequality which differs for 
each only by its coefficients, and which forms in these motions a sort of 
libration whose extent is arbitrary. Observations show it to be insen 
sible. 

53 1. Let us now consider the variations of the excentricities and of the 
perihelions of the orbits. For this purpose, resume the expressions of 
d f, d F, d f" found in 53 T : calling the radius-vector of /* projected 
upon the plane of x, y ; v the angle which this projection makes with the 
axis of x ; and s the tangent of the latitude of <A above the same plane, we 
shall have 

x = P cos. v ; y = sin. v ; z = g s 

whence it is easy to obtain 



d R N /d Rx , 2X /d 

x - z = ] + 8 > cos - v ~ s cos - v 



d R 

s S1U - v 



d R 

s sin - v -d7 - s sm - v 

/d Rx 

- S COS. V ( -j- 

\d v / 

By 531, we also have 

xdy ydxrrcdt; xdz zdxrrc dt; ydz zdy = c dt; 

M2 



ISO A COMMENTARY ON [SECT. XL 

the differential equations in f, P, f " will thus become 



df = - d y -* < +s )co,v ()_, i co.. v ( 

/ dH \ I 
+ s sm - v (ar) / 



dRx) 



L^ 

\d s / J \d s / 



, ,. . d R 
d y (1 + s 2 ) sin. 



d f / |Vdv g s 

d R\ cos. v /d R\ s. sin. v /d 



, . 
. d t sm. v 

The quantities c , c" depend, as we have seen in No. 531, upon the in 
clination of the orbit of # to the fixed plane, in such a manner that they 
become zero when the inclination = ; moreover it is easy to see by the 

nature of R that (, ) is of the order of the inclinations of the orbits ; 
v. d s/ 

neglecting therefore the squares and products of these inclinations, the 
preceding expressions of d f and of d f , will become 

, ,. i /d R\ j f /d R\ . cos. v /d R 

d f = ~- - d - - c d t ^ sm - v 



. , /d R\ , / /d R\ sin. v /d R\ \ 

f = d x (av) + c d l l cos - v C-di) - - (ar, )S - 



but we have 

d x = d (g cos. v) ; d y = d (g sin. v); cdt=xdy ydx = g s dv, 

we therefore get 

4 f = [d s sin. v + 2 g d v cos. v} (^ ) f 2 d v sin. v (-T- ) j 

d f = Jd g cos. v 2 f d v sin. v} (-j ) + ? s d v cos. v (-^ ). 

These equations are more exact, if we take for the fixed plane of x, y, 



BOOK I.] NEWTON S PRINC1PIA. 181 

that of the orbit of p, at a given epoch ; for then c , c" and s are of the 
order of the perturbing forces ; thus the quantities which we neglect, are 
of the order of the squares of the perturbing forces, multiplied by the 
square of the respective inclination of the two orbits of p and of /& . 

The values off, d f, d v, (-^ V (-, \ remain clearly the same what 
ever is the position of the point from which we reckon the longitudes ; 
but in diminishing v by a right angle, sin. v becomes cos. v, and cos. v 
becomes sin. v ; the expression of d f changes consequently to that of 
d f ; whence it follows that having developed, into a series of sines and 
cosines of angles increasing proportionally with the times, the value of 
d f, we shall have the value of d f , by diminishing in the first the angles 
i, i , *, , 6 and 6 by a right angle. 

The quantities f and f determine the position of the perihelion, and 
the excentricity of the orbit ; in fact we learn from 531, that 

f 
tan. 1 = r ; 

I being the longitude of the perihelion referred to the fixed plane. When 
this plane is that of the primitive orbit of ^, we have up to quantities of 
the order of the squares of the perturbing forces multiplied by the square 
of the respective inclinations of the orbits, I &, -a being the longitude of 
the perihelion upon the orbit ; we shall therefore then have 

P 



tan. . 



which gives 



cos. *r =r 



V f- + f /2 V f* + f 2 

By 531, we then get 

f / c > f c /> 

/ .1*9 i L / 9 i i.*// 9 (*lt *"* * V" 

me = V i 2 + r 2 + i z , f " = : 

c 

thus c and c" being in the preceding supposition of the order of the 
perturbing forces, f" is of the same order, and neglecting the terms of the 
square of these forces, we have 
m e = V f + f /2 . 



If we substitute for V f 2 + f *, its value m e, in the expressions of 
sin. w, and of cos. w, we shall have 

m e sin. = f ; me cos. w = f ; 

these two equations will determine the excentricity and the position of the 
perihelion, and we thence easily obtain 

m z . e d e = f d f + f d f ; m 2 e d = f d f f d f. 

M3 



182 A COMMENTARY ON [SECT. XI. 

Taking for the plane of x, y that of the orbit of /A; we have for the 
cases of the invariable ellipses, 

- a (1 e 2 ) . _ g fc d v . e . sin, (v r) _ 

s " 1 + e cos. (v tr) S ~ a(l e 2 ) 

g 2 d v = a 2 n d t VI e 2 ; 

and by No. 530, these equations also subsist in the case of the variable 
ellipses ; the expressions of d f and of d f will thus become 

d f = _ -==- 2 cos. v + | e cos. +\ e cos. (2 v *)} 



- a n d t V 1 e 2 . sin. v . 



d 



df> = -- andt [2 sin. v+| e sin. <*+ e sin. (2 v r)J 
VI e 2 

+ a 2 n d t V 1 e 2 . cos. v(-r ); 



wherefore 

andt . N c ,. ,, X7 /d R\ 

e d * = -- 7T=1 sin (v r) [2 + e cos. (v *)} f-r- ) 

2 



m V 1 e 

, / \ 

. (v w\ ( -: ) 
\ d / 



a 2 , n d t V 1 e 2 , /d R 

- cos. 



m 



e = 



m V I e 



.\ 



m 



This expression of d e may be put into a more commodious form in 
some circumstances. For that purpose, we shall observe that 



substituting for g and d their preceding values, we shall have 



but we have 

P * d v = a 2 n d t V 1 e 2 ; 

n d t [I + e cos. (v *)}* m 
d v = s > 

(1-e 2 ) ^ 

wherefore 

,-i i / x /d R 

a 2 ndt V 1 e~. sin. (v -) 



e V 1 e" 



BOOK 1.] NEWTON S PRINCIPIA. 183 

the preceding expression of d e, will thus give 

a n d t V 1 e 2 /d Rx a (1 e 2 ) 

p H p - - . I = - I - - U IV. 



m v/ m 

We can arrive very simply at this formula, in the following manner 
We have by No. 531, 

d c /d Rx /d Rx /d l 



but by the same No. c = V m a (1 e 2 ) which gives 

d a V m a (1 -^ e s ) e d e V m a 
d c = - ^ s --- . , ; 
2 a VI e z 

therefore 

da 



m Vdv; 2a 2 

and then we have by No. 53 1 

^ = - d R. 
2 a 2 

We thus obtain for e d e the same expression as before. 

535. We have seen in 532, that if we neglect the squares of the per 
turbing forces, the variations of the principal axis and of the mean mo 
tion contain only periodic quantities, depending on the configuration of 
the bodies /*, //, ^", &c. This is not the case with respect to the varia 
tions of the excentricities and inclinations : their differential expressions 
contain terms independent of this configuration and which, if they were 
rigorously constant, would produce by integration, terms proportional to 
the time, which at length would render the orbits very excentric and 
greatly inclined to one another ; thus the preceding approximations, found 
ed upon the smallness of the excentricity and inclination of the orbits, 
would become insufficient and even faulty. But the terms apparently 
constant, which enter the differential expressions of the excentricities and 
inclinations, are functions of the elements of the orbits ; so that they vary 
with an extreme slowness, because of the changes they there introduce. 
We conceive there ought to result in these elements, considerable inequa 
lities independent of the mutual configuration of the bodies of the system, 
and whose periods depend upon the ratios of the masses y.+ /a, , &c. to the 
mass M. These inequalities are those which we have named secular in 
equalities, and which have been considered in (520). To determine them 
by this method we resume the value of d f of the preceding No. 

d f = {2 cos v + I e cos. * 4- A e cos. (2 v *)} [, ) 

VI e 2 Wl v 



184 A COMMENTARY ON [SECT. XI. 



a 2 n d t V 1 e 2 .sin 



d 

We shall neglect in the developement of this equation the square and 
products of the excentricities and inclinations of the orbits ; and amongst 
the terms depending upon the excentricities and inclinations, we shall re 
tain those only which are constant : we shall then suppose, as in No. 515. 
S = a(l + u,); / = a (l + u/) ; 
v = n t + -f v, ; v = n 7 t -f s + v/. 

Again, if we substitute for R, its value found in 515; if we next con 
sider that by the same No. we have, 

d Rx a /d Rx /d 



and lastly if we substitute for u /5 u/, v /} v/ their values e cos. (n t+ 1 r), 
e cos. (n t + t ), 2 e sin. (n t + t *), 2 e sin. (n f t + ) 
given in No. 484, &c. by retaining only the constant terms of those which 
depend upon the first power of the excentricities of the orbits, and ne 
glecting the squares of the excentricities and inclinations, we shall find 
that 



a ^ n d t. 5 j i A + 1 a (^ ) } sin. Ji(n 1 n t + e - s) + n t + *}; 

the integral sign belonging as in the value of R of 515, to all the whole 
positive and negative values of i, including also the value of i = 0. 

We shall have by the preceding No. the value of d f, by diminishing 
in that of d f the angles i, , *, =/ by a right angle; whence we get 

a (j! n d t ( /d A v . /d 2 A <> 

- . . e. cos. ~ a __ a* - 



( 

. e. cos. ~ ja 



,. mi 
-a/ndt e . cosV A 0) + i 



r /d A ^ \ i 

+ a/Vndt. 2-j iA (i) + ^a {--; - J Vcos.i (n- t n t+j s) + n t+s]. 

Let X, for the greater brevity, denote that part of d f, which is con 
tained under the sign 2, and Y the corresponding part of d i . Make also, 
as in No. 522, 

nn * /n / 

1} = ~-r | a 



BOOK I.] NEWTON S PRINCIPIA. 18.5 



then observe that the coefficient of e d t sin. <JT , in the expression of d f, 
is reducible to |0, Ij when we substitute for the partial differences in a , 
their values in partial differences relative to a; finally suppose, as in 517, 
that 

e sin. zt h ; e sin. = h 

e cos. -or = 1 ; e cos. / = \ f 

which gives by the preceding No. f =r m 1, f = m h or simply f = I, 
hj by taking M for the mass-unit, and neglecting & with regard to 
M ; we shall obtain 

j= (0, l).l-joTT.l +aA* nY; 



< = - (0, 1). h + |0, 1|. h - a yf n. X. 

Hence, it is easy to conclude that if we name (Y) the sum of the terms 
analogous to a /* n Y, due to the motion of each of the bodies fj. , p", &c. 
upon ^ ; that if we name in like manner (X) the sum of the terms analo 
gous to a fjt/ n X due to the same actions, and finally if we mark suc 
cessively with one dash, two dashes, &c. what the quantities (X), (Y), h, 
and 1 become relatively to the bodies fjf, A", &c. ; we shall have the fol 
lowing differential equations, 
dh 



= 1(0,1) + (0,2) + &c.} 1 - [0,J 1 - JOTS) 1" - &c. + (Y); 
~ = J(0, 1) + (0, 2) + &c.J h + OH] h + |OT2| h" + &c+ (X) ; 
| = {(1, 0) + (1, 2) + &c.} 1 _ [T70| 1 - [172] 1" - & c . + (Y ) 

~ - - J(l, 0) + (1, 2) + &c.} h + O h + [iT2|h^+&c.+ (X / ) 

&c. 

To integrate these equations, we shall observe that each of the quanti 
ties h, 1, h , F, &c. consists of two parts ; the one depending upon the 
mutual configuration of the bodies , //, &c. ; the other independent of 
this configuration, and which contains the secular variations of these quan 
tities. We shall obtain the first part by considering that if we regard 
hat alone, h, 1, h , 1 , &c. are of the order of the perturbing masses, and 
consequently, (0, 1). h, (0, 1). 1, &c. are of the order of the squares of 



186 A COMMENTARY ON [SECT. XL 

these masses. By neglecting therefore quantities of this order, we shali 
nave 

d n _. /v\ . d * -. cv \ . 

dT - (Y) dT 

dh; __ m . dj[ _ t 

d t " v d t " 
wherefore, 

h=/(Y)dt; l=/(X)dt; h =/(Y )dt; &c. 

If we take these integrals, not considering the variability of the ele 
ments of the orbits and name Q what/(Y) d t becomes ; by calling 3 Q 
the variation of Q due to that of the elements we shall have 

/(Y)dt = Q-/5Q; 

but Q being of the order of the perturbing masses, and the variations of 
the elements of the orbits being of the same order, 5 Q is of the order of 
the squares of the masses ; thus, neglecting quantities of this order, we 
shall have 

/(Y) d t = Q. 

We may, therefore, take the integrals/ (Y) d t, / (X) d t, / (Y ) d t, 
&c. by supposing the elements of the orbits constant, and afterwards con 
sider the elements variable in the integrals ; we shall after a very simple 
method, obtain the periodic portions of the expressions of h, 1, h , &c. 

To get those parts of the expressions which contain the secular inequa 
lities, we observe that they are given by the integration of the preceding 
differential equations deprived of their last terms, (Y), (X), &c. ; for it is 
clear that the substitution of the periodic parts of h, 1, h , &c. will cause 
these terms to disappear. But in taking away from the equations their 
last terms, they will become the same as those of (A) of No. 522, which 
we have already considered at great length. 

536. We have observed in No. 532, that if the mean motions n t and 
n t of the two bodies & and X are very nearly in the ratio of i to i so 
that V n in may be a very small quantity ; there may result in the 
mean motions of these bodies very sensible inequalities. This relation of 
the mean motions may also produce sensible variations in the excentrici- 
ties of the orbits, and in the positions of their perihelions. To determine 
them, we shall resume the equation found in 534, 

an dt. VI e 2 /d R\ a (1 e 2 ) 7 _, 

e d e = . ( -r ) S - d R. 

m \ d v / m 

It results from what has been asserted in 515, that if we take for the 
fixed plane, that of the orbit of /*, at a given epoch, which allows us to 



BOOK I.] NEWTON S PRINCIPIA. 187 

neglect in R the inclination <p of the orbit of ^ to this plane; all the terms 
of the expression of R depending upon the angle i n t i n t, will be 
comprised in the following form, 

li! k cos. (i n t i n t + i i t g * g / J g ^), 
i, i , g, g, g" being whole numbers and such that we have = i -i-g-g -g". 
The coefficient k has the factor e . e * (tan. <p ) *" ; g, g , g" being taken 
positively in the exponents : moreover, if we suppose i and V positive, and 
i greater than i; we have seen in No. 515, that the terms of R which 
depend upon the angle i n t i n t are of the order i i, or of a su 
perior order of two, of four, &c. units ; taking into account therefore only 
terms of the order i i, k will be of the form e . e * (tan. |- <ff) *". Q, 
Q being a function independent of the excentricities and the inclination 
of the orbits. The numbers g, g , g" comprehended under the symbol 
cos., are then positive ; for if one of them, g for instance, be negative and 
equal to f, k will be of the order f + g + g" ; but the equation = i 
i g g g" gives f + g + g" = i i + 2 f ; thus k will be 
of an order superior to i i, which is contrary to the supposition. Hence 

J T> J T> 

by No. 515, we have ( -, ) = (, ) provided that in this last partial 
J \dv/\d/ 

difference, we make t -a constant; the term of (-* j corresponding 

to the preceding term of R, is therefore 

/ (i + g) k sin. (i n t i n t + i i e g g J g" 6 ). 

The corresponding term of d R is 
, i n k d t sin. (i n t i n t + V t i e g =r g * g" & }. 

Hence only regarding these terms and neglecting e 2 in comparison with 
unity, the preceding expression of e d e, will give 

ul a n d t Q k . . , 

d e = . 2 sm. (i n t i n t -f- i s i t g g */ g" r) , 

but we have 

ge- . e * . (tan. |p ) g// . Q= (^ 



integrating therefore we get 

e = P7 . . . ( -.- ^ cos. (i n t 5 n t + i t i i g ^ g o- g" ^). 

m (i n in) \d e/ 

The sum of all the terms of R, however, which depend on the angle 
i n t i n t. being represented by the following quantity 
/<* . P sin. (i n t i n t + i E i + tt/ P cos. (i n t i n t + i i t) 
the corresponding part of e will be 



188 A COMMENTARY ON [SECT. XI. 

This inequality may become very sensible, if the coefficient i n i n 
is very small, for it actually takes place in the theory of Jupiter and Sa 
turn. In fact, it has for a divisor only the first power of i n i n, whilst 
the corresponding inequality of the mean motion, has for a divisor the se 
cond power of this quantity, as we see in No. 532; butf 1 ) and f -, ) 

being of an order inferior to P and P , the inequality of the excentricity 
may be considerable, and even surpass that of the mean motion, if the 
excentricities e and e are very small ; this will be exemplified in the 
theory of Jupiter s satellites. 

Let us now determine this corresponding inequality of the motion of 
the perihelion. For that purpose, resume the two equations 

fdf+f df fdf f df 

ede = - -^- -, e d^--^ - 

which we found in No. 534. These equations give 

d f =r m d e cos. -a m e d . sin. ~; 
thus with regard only to the angle 

i n t i n t + i e is g z, g * g" 6 , 
we shall have 

d f = (if. a n d t (^) cos. sin. (i n t i n t + i i g ~ g g"0 

m e d -a . sin. &. 
Representing by 

ii/. a n d t { (^) + k } cos. (i n t i n t + i * i g ~ g g" 8), 
the part of m e d , which depends upon the same angle, we shall have 
d f = (ii. a n d t { (^) + |k } sin.(i n t-i n t + i f -i (g-l)*-gV-g"O 

^"-^k sin. (i n t int + iV it (g + l) g */ g Y). 



It is easy to see by the last of the expressions of d f, given in the No. 
534, that the coefficient of this last sine has the factor e e + l . e g/ (tan. \ p) g " ; 

k is therefore of an order superior to that of (ir 1 -) by two units; thus, 

(cl lc\ 
-j J , we shall have 

.andt /d k\ ... . , / / / n" a\ 

cos. (i n 7 1 i nt+iV it g g- g 7 ff) 



m \d e 

for the term of e d w, which corresponds to the term 
(jj k cos. i n t \nt + \ i it * 



BOOK I.] NEWTON S PRINCIPIA. 189 

of the expression of R. Hence it follows that the part of w, which cor 
responds to the part of R expressed by 

a, P sin. (i n t i n t + i i t} + (jJ P cos. (Vn t int+Vt i e), 
is equal to 

r--r-r-^\- } ( i ^ cos.(i n t-int + i s -ii}- (, ^ sin.(i n t-int + rY-ig) c > 
m(i n -in)e t\de/ V d e / 

we shall therefore, thus, after a very simple manner, find the variations 
of the excentricity and of the perihelion, depending upon the angle 
i n t i n t + i e i e. They are connected with the variation oi 
the corresponding mean motion, in such a way that the variation of the 
excentricity is 



3in Vde.dt 
and the variation of the longitude of the perihelion is 
i n in /d A 

Sine \d~e) 

The corresponding variation of the excentricity of the orbit of //, due 
to the action of ^ will be 

_!_ fj-\ 

3i n . e Vde .d J 

and the variation of the longitude of its perihelion, will be 
i n in /d 



3 i n e \< 

and since by No. 532, = ^ a , . , the variations will be 

fj> v a 

i* V a. / d 2 g \ , (i n i n) /, V a d 
3 i . n . y! V a VdVTd t) a "srii^V V^a dV 

When the quantity i 7 n i n is very small, the inequality depending 
upon the angle i n t i n t, produces a sensible one in the expression 
of the mean motion, amongst the terms depending on the squares of the 
perturbing masses ; we have given the analysis of this in No. 532. This 
same inequality produces in the expression of d e and of d =r, terms of 
the order of the squares of the masses, and which, being only functions of 
the elements of the orbits, have a sensible influence upon the secular 
variations of these elements. Let us consider, in fact, the expression of 
d e, depending on the angle i n t in t. 

By what precedes, we have 



de = 



/* . a n . d t 



m 



Ud P\ ,.. . 

-T ] cos. (r n t i n t + i % is) 



" ("d~e") Sin< ( l/ t - i n t + i - 



190 A COMMENTARY ON [SECT. XI. 

By No. 532 the mean motion n t, ought to be augmented by 

-r ? a " I 1 - \ Pcos. (i n t int+iV is) Fsin.(i n t-i n t+ i i i ) I 
(in in) 2 .m I ; J 

and the mean motion n t, ought to be augmented by 

3 fjf a n 2 . i /* V a fri .., . . . , ., . . . 
7*7 F ! -n / / /-P cos. (i n t i n t + i t i 6 ) 
(i n in) 2 . m ^ V a 

F sin. (i n t i n t + i e i )}. 

In virtue of these augments, the value of d e will be augmented by the 
function 

3^a 2 . in 3 , dt ,. , . , , ., , 7 f /dP x , /dP \ 1 
i./^ V a +i> v aM P. f-. ) + P f-i ) J-; 
I ^de/ \de/J 



/ / // / \ 
a . (I n -in) 

and the value of d w will be augmented by the function 

3 (if a 2 . i n 3 . d t . . / D /d PN , _, /d P\ ) 

5 8 7 /// / - S-TT^ fi^ Va + i> Va}. IP. (j ) + Fl-TJ f . 
3m 1 v (r nr in) 2 , e I \d e / vde/J 

In like manner we find that the value of d e will be augmented by the 
function 



and that the value of d e will be augmented by the function 

d 



These different terms are sensible in the theory of Jupiter and Saturn, and 
in that of Jupiter s satellites. The variations of e, e , &, *r relative to the 
angle i n t i n t may also introduce some constant terms of the order of 
the square of the perturbing masses in the differentials d e, d e , dw, and d*/, 
and depending on the variations of e, e , w, & relative to the same angle. 
This may easily be discussed by the preceding analysis. Finally it will 
be easy, by our analysis, to determine the terms of the expressions of 
e, , e , w which depending upon the angle i n t i n t + \ f * i e 
have not i n in for a divisor, and those which, depending on the same 
angle and the double of this angle, are of the order of the square of the 
perturbing forces. These different terms are sufficiently considerable in 
the theory of Jupiter and Saturn, for us to notice them : we shall deve- 
lope them to the extent they merit when we come to that theory. 

537. Let us determine the variations of the nodes and inclinations of 
the orbits, and for that purpose resume the equations of 53 1 , 



BOOK I.] NEWTON S PRINCIPIA. 191 

, R 

dc = 

, 

= 



If we only notice the action of /i* 7 , the value of R of No. 513, gives 



d R\ /d R 

" x 



RN /d R 



Let however, 



c 



the two variables p and q will determine, by No. 53 1, the tangent of the 
inclination <p of the orbit of /*, and the longitude 6 of its node by means of 
the equations 



tan. <p = V p 2 -f q 2 ; tan. d = -_ . 

Call p 7 , q , p", q", &c. what p and q become relatively to the bodies 
/A 7 , At", &c. : we shall have by 531, 

z = q y p x ; z = q y p 7 x , &c. 
The preceding value of p differentiated gives 
d p J_ d c /7 p d c 
dt :: T dt 

substituting for d c, and d c 77 their values we get 

af = Kq q ) y / + (P - P) x/ y? x 

Hx l + y 2 + z 1 ) 1 J( X 7 _x) + (y _ y )*-f( z <--z) 
In like manner we find 

= ^ (P/ - P) x x 7 + (q _ q 7 ) x y 7 } X 



192 A COMMENTARY ON SECT. XL 



3 

x 2 + y" +y*) KX x )*+(y _ y) 2 +(z z) 2 } 

If we substitute for x, y, x , y their values g cos. v, o sin. v, % cos. v , 
g sin. v , we shall have 

(q q ) y y + (p p) * y = q J3 - s ? i cos - (v +v) cos. (v v)i 



sn - 



(p p) x x + (q q ) x y = j^- S $ i*- (v +v) + cos. (v v)} 

+ ^^^ g ? {sin. (v +v) + sin. (v v)}. 

Neglecting the excentricities and inclinations of the orbits, v, e have 

s = a ; v = n t + f ; ? = a ; v = n t + ; 
which give 

_ 1 __ ___ 1 _ _ 1_ 
(x /s + y /f + z /2 )* Ux x) s + (y y)*+ ( Z _ Z )^f ~a /3 



a 2 2 a a cos. (n t n t + s) + a /s ] 
moreover by No. 516, 

- ? - 5- = \ 2. B . cos. i (n 7 1 n t+* ) 
{a 2 2 a a cos. (n t n t + *) +a 2 }^ 

the integral sign 2 belonging to all whole positive and negative values of 
i, including the value i = ; we shall thus have, neglecting terms of the 
order of the squares and products of the excentricities and inclinations of 
the orbits, 



dp q m _ 

<& c a 

. jsin. ( n t+ nt+ +) sin. (n t nt + . 



m _ . {coSm (n / 1 + n t + ,/ + g) _ cos . (n / 1 _ n t+ ,/_, 

cl t <& c a 



c a 

- q . /. a a . 2. B cos.[(i+ 1) (n t n t+i 7 0] 

C 

cos.[(i+l) (n t n t+e e) + 2nt+2*]} 
=^ . /. a a . 2. B W fsin.[(i+l) (iV t n t+a i)J 

C 

sin.[(i+l) (n t n t+i 0+ 2nt+2]|. 

* Jcos> (n/ 1 + n l + * + ) + cos< (11/ 1- " n t+s/ ~ ^ 



BOOK I.] NEWTON S PRINCIPIA. 193 

+ c l _9. ^ . [sin. (n t + n t + s +e) + sin. (n t--nt + i / i)} 

wl C U 

+ p -p2- . v!. a a . 2. B W.{cos. [(i+ 1) (n t n t-H 0] 

TP C 



+ cos. [(i+1) (n t n t+ i f t) + 2 
+ 2p3. ,(* . a a . 2. B W. sin. [(i+1) (n t n t+i )] 

T0 O 

+ sin. [(i + 1) (n t n t+/ s) + 2 n t+2 OJ- 
The value i = 1 gives in the expression of - , the constant quan 

tity -~ - . /* . a a B ( !) : all the other terms of the expression of -~~ 
4 c d t 

are periodic : denoting their sum by P, and observing that B ( ~ !) = B W 
by 516, we shall have 

i? = i.=L3. A* . a a . B<" + P. 
at 4 c 

By the same process we shall find, that if we denote by Q the sum of 
all the periodic terms of the expression of-r-j" , we shall have 

U L 



.. . 

d t 4 c 

If we neglect the squares of the excentricities and inclinations of the 
orbits, by 531, we have c r= V m a, and then supposing m = 1, we 

have n 2 a 3 = 1 which gives c = ; the quantity f/ " a a " - thus be- 

an 4 c 

comes - ^ - which by 526, is equal to (0, 1); hence we get 
lH = (0, 1). (q -q)+P; 

^ = (0, 1). (p _ p ) + Q. 

Hence it follows that, if we denote by (P) and (Q) the sum of all the 
functions P and Q relative to the action of the different bodies fjf t p", &c. 
upon A*; if in like manner we denote by (P), (Q ), (P"), (Q"), &c. what 
(P) and (Q) become when we change successively the quantities relative 
to p into those which are relative to /, /A", &c. and reciprocally ; we shall 
have for determining the variables p, q, p , q , p", q", &c. the following 
system of differential equations, 

^ P t = {(0, 1) + (0, 2) + &c.} q + (0, 1). q + (U, 2) q"+ &c.+ (P) ; 



VOL. II. N 



194 A COMMENTARY ON [SECT. XI. 

jS. ={(0, 1) + (0, 2) + &c.} p (0, 1) p - (0, 2) p" - &c. + (Q) ; 



= {(1, 0) + (1, 2) + &c.J q + (1, 0) q + (1, 2) q"+ &c. + (F); 
L ={(],0) + (1,2) + &c^p -(l,0)p-(l,2)p"-&c. 



&c. 

The analysis of 535, gives for the periodic parts of p, q, p , q , &c. 
p =/(P).dt; q =/(Q).dt; 
p =/(F).dt; q =/(Q ).dt; 
&c. 

We shall then have the secular parts of the same quantities, by inte 
grating the preceding differential equations deprived of their last terms 
(P), (Q), (P ), &c. ; and then we shall again hit upon the equations (C) 
of No. 526, which have been sufficiently treated of already to render it un 
necessary again to discuss them. 

538. Let us resume the equations of No. 531, 

V c 2 + c" 1 c" 

tan. p = - ; tan. Q = 
c c 

ivhence result these 

c c" 

= tan. cos. 6 ; - = tan. sin. i. 
c c 

Differentiating, we shall have 

d tan. p = {d c cos. & + d c" sin. 6 d c tan. <p] 

C 

d 6 tan. <p = - {d c" cos. 6 d c sin. 6}. 

C 

If we substitute in these equations for -y , -y , -r , their values 

/d Rx /d Rx /d Rx /d Rx /d Rx /d Rx , e 
V ( T ) x [ T ) , z I T ) xl -, ) , z ( -j } y ( i ) , and for 
J Vdx/ >-dy/ \dx/ vdz/ Vd y / Viz/ 

these last quantities their values given in 534 ; if moreover we observe 
that s = tan. <p sin. (v 0), we shall have 

d t tan. cos. (v - 6) f /d Rx . , ., , /d R 
_ __ 



. tan. p = 

1 + s 2 dt . /d R 



,. d t tan. sin. (v - 6) ( /d Rx . . . /d R 

d 6 . tan. p = - -^- - | ? . ( d - ) sm.(v-^)+ (^- 

(1 + s 2 ) dt . . .,/d Rx 
- ! - L - sin . (v tfH-T J. 
c \ d s / 



BOOK I.] NEWTON S PRINCIPI A. 19 i 

These two differential equations will determine directly the inclination 
of the orbit and the motion of the nodes. 

They give 

gin. (v 0} d tan. <p d 6 cos. (v 6} tan. <p = 0; 
an equation which may be deduced from this 

s = tan. <p sin. (v 6} ; 

in fact, this last equation being finite, we may (530) differentiate it whe 
ther we consider <f> and d constant or variable ; so that its differential, 
taken by only making <p and d vary, is nothing ; whence results the pre 
ceding differential equation. 

Suppose, however, that the fixed plane is inclined extremely little to the 
orbit of /a, so that we may neglect the squares of s and tan. f>, we shall 
have 



, . t . . /d R\ 

d 6 tan. <p = --- sin. (v 6} IT ); 
c \ds J 

by making therefore as before 

p = tan. p sin. & ; q = tan. <p cos. 6 ; 

we shall have, instead of the preceding differential equations, the follow 
ing ones, 

d t /d Rx 

d q = --- cos. v . ( -T ) ; 
c \ d s / 

d t . /d Rx 
d p = -- sin. v . ( -j ) ; 
c \ d s / 

But we have also 

s = q sin. v p cos. v 
which gives 

/dRx _ I /d Rx /d JRx _ \ /d Rx 

\ds/ sin. v vdq/ \ds/~ cos. v \dp/ 
wherefore 

d t d 



d t/d 



We have seen in 515 that the function R is independent of the po 
sition of the fixed plane of x, y ; supposing, therefore, all the angles 01 
that function referred to the orbit of //-, it is evident that R will be a 

function of these angles and the respective inclination of two orbits, an 

N2 



196 A COMMENTARY ON [SECT. XI. 

inclination we denote by p/. Let 6J be the longitude of the node of the 
orbit of /jf upon the orbit of/*; and supposing that 

ti! k (tan. p/) cos. (i n t i n t + A g 6f) 

is a term of R depending on the angle i n t i n t, we shall have, by 
527, 

tan. p/ . sin. 6f = p p ; tan. p/ cos. 6/ = q q ; 
whence we get 



(tan. p/) sin. g /= iq -q + (p - P ) V-H -fr -q-(p -p) V- 



(tan. ,/) . COS. g /= - - --q- (p -p) V-H 

With respect to the preceding term of R, we shall have 

(tip") = S ( tan< ?/) TV k sin - H n t in t + A (g 1) 0/J ; 

= ~ g (tan * P /)8 ~ V k cos> ** n ~ * n l + A (S ] ) /! 



If we substitute these values in the preceding expressions of d p and 
d q, and observe that very nearly c = , we shall have 



Substituting these values in the equation 

s = q sin. v p cos. v 
we shall have 



s= -- g * * * a n 
m 



* ; , * (tan. ? /)- sin. f i n t i n t v + A (g -1) 6f}. 
(in i n) v 

This expression of s is the variation of the latitude corresponding to 
the preceding term of R : it is evident that it is the same whatever may 
be the fixed plane to which we refer the motions of ^ and /V, provided that 
it is but little inclined to the plane of the orbits ; we shall therefore thus 
have that part of the expression of the latitude, which the smallness of the 
divisor i n in may make sensible. Indeed the inequality of the lati 
tude, containing only the first power of this divisor, is in that degree 
less sensible than the corresponding inequality of the mean longitude, 
which contains the square of the same divisor ; but, on the other hand, 
tan. <pf is then raised to a power less by one ; a remark analogous to that 
which was made in No. 536, upon the corresponding inequality of the 
excentricities of the orbits. We thus see that all these inequalities are 



BOCK I.] NEWTON S PRINCIPIA. 197 

connected with one another, and with the corresponding part of R, by 
very simple relations. 

If we differentiate the preceding expressions of p and q, and if in the 

values of -a-*- and ^ - we augment the angles n t and n t by the inequa 

lities of the mean motions, depending on the angle i n t i n t, there 
will result in these differentials, quantities which are functions only of the 
elements of the orbits, and which may influence, in a sensible manner, the 
secular variations of the inclinations and nodes although of the order of 
the squares of the masses. This is analogous to what was advanced in 
No. 536 upon the secular variations of the excentricities and aphelions. 

539. It remains to consider the variation of the longitude t of the epoch. 
By No. 531 we have 



d ^ { E n cos. ( v w ) + E - cos. 2 ( v w) + &c.] ; 
substituting for E p , E ;V \ &c. their values in series ordered according to 

O * J vJ 

the powers of e, series which it is easy to form from the general expres 

sion of E : } (473) we shall have 

d i = 2 d e sin. (v *) + 2 e d w cos. (v &) 

+ e d e \l + \ e 2 +&c.} sin. 2 (v ) e 2 d {f + e 2 + &c.}cos.2 (v ~) 

e 2 d e U + &c.} sin. 3 (v ) + e 3 d U + &c.} cos. 3 (v *r) 

+ &c. 

If we substitute for d e and e d * their values given in 534, we shall 
find, carrying the approximation to quantities of the order e * inclusively, 

de = a2 nd Vl eM2 fecos. (v ) + e 2 cos. 2 (v 
in 

a n d t . . . r, , x, /d R 



. . . r, , x, / 

. e . Sin. (V - nr) \ 1 + i e COS. (V - at}\ [ 

^ 



m V 1 e 2 
The general expression of d t contains terms of the form 

X k . n d t . cos. (i n t i n t + A) 
and consequently the expression of i contains terms of the form 

T-. ; - -. sin. (i n t i n t + A) ; 
in i n 

but it is easy to be convinced that the coefficient k in these terms is of 
the order i i, and that therefore these terms are of the same order as 
those of the mean longitude, which depend upon the same angle. These 
having the divisor (i n in) *, we see that we may neglect the corre 
sponding terms of f, when i n i n is a very small quantity. 

N3 



198 A COMMENTARY ON [SECT. XL 

If in the terms of the expression of d e, which are solely functions of the 
elements of the orbits, we substitute for these elements the secular parts 
of their values ; it is evident that there will result constant terms, and 
others affected with the sines and cosines of angles, upon which depend 
the secular variations of the excentricities and inclinations of the orbits. 
The constant terms will produce, in the expression of E, terms propor 
tional to the time, and which will merge into the mean motion p. As to 
the terms affected with sines and cosines, they will acquire by integration, 
in the expression of s, very small divisors of the same order as the per 
turbing forces ; so that these terms being at the same time multiplied and 
divided by the forces, may become sensible, although of the order of the 
squares and products of the excentricities and inclinations. We shall see 
in the theory of the planets, that these terms are there insensible; but in 
the theory of the moon and of the satellites of Jupiter, they are very sen 
sible, and upon them depend the secular equations. 

We have seen in No. 532, that the mean motion of/,*, is expressed by 

--//andt.rfR, 

and that if we retain only the first power of the perturbing masses, d R 
will contain none but periodic quantities. But if we consider the squares 
arid products of the masses, this differential may contain terms which are 
functions only of the elements of the orbits. Substituting for the elements 
the secular parts of their values, there will thence result terms affected with 
sines and cosines of angles depending upon the secular variations of the 
orbits. These terms will acquire, by the double integration, in the ex 
pression of the mean motion, small divisors, which will be of the order of 
the squares and products of the perturbing masses; so that being both 
multiplied and divided by the squares and products of the masses, they 
become sensible, although of the order of the squares and products of the 
excentricities and inclinations of the orbits. We shall see that these terms 
are insensible in the theory of the planets. 

540. The elements of p s orbit being determined by what precedes, by 
substituting them in the expressions of the radius-vector, of the longitude 
and latitude which we have given in 484, we shall get the values of these 
three variables, by means of which astronomers determine the position of 
the celestial bodies. Then reducing them into series of sines and cosines, 
we shall have a series of inequalities, whence tables being formed, we may 
easily calculate the position of ^ at any given instant. 

This method, founded on the variation of the parameters, is very useful 



BOOK I.] NEWTON S PKINCIPIA. 199 

in the research of inequalities, which, by the relations of the mean motions 
of the bodies of the system, will acquire great divisors, and thence become 
very sensible. This sort of inequality principally affects the elliptic ele 
ments of the orbits ; determining, therefore, the variations which result 
in these elements, and substituting them in the expression of elliptic mo 
tion, we shall obtain, in the simplest manner, all the inequalities made 
sensible by these divisors. 

The preceding method is moreover useful in the theory of the comets. 
We perceive these stars in but a very small part of their courses, and ob 
servations only give that part of the ellipse which coincides with the arc 
of the orbit described during their apparitions ; thus, in determining the 
nature of the orbit considered a variable ellipse, we shall see the changes 
undergone by this ellipse in the interval between two consecutive appari 
tions of the same comet. We may therefore announce its return, and 
when it reappears, compare theory with observation. 

Having given the methods and formulas for determining, by successive 
approximations, the motions of the centers of gravity of the celestial bo 
dies, we have yet U) apply them to the different bodies of the solar system : 
but the ellipticity of these bodies having a sensible influence upon the 
motions of many of them, before we come to numerical applications, we 
must treat of the figure of the celestial bodies, the consideration of which 
is as interesting in itself as that of their motions. 



SUPPLEMENT 



TO 



SECTIONS XII. AND XIII. 

ON ATTRACTIONS AND THE FIGURE OF THE CELESTIAL BODIES. 

541. The figure of the celestial bodies depends upon the law of gravi 
tation at their surface, and the gravitation itself being the result of the at 
tractions of all their parts, depends upon their figm e; the law of gravi 
ty at the surface of the celestial bodies, and their figure have, therefore, a 
reciprocal connexion, which renders the knowledge of the one necessary 
to the determination of the other. The research is thus very intricate^ 

N4 



200 A COMMENTARY ON [SECT. XII. & XIII. 

and seems to require a very particular sort of analysis. If the planets were 
entirely solid, they might have any figure whatever ; but if, like the earth, 
they are covered with a fluid, all the parts of this fluid ought to be dis 
posed so as to be in equilibrium, and the figure of its exterior surface de 
pends upon that of the fluid which covers it, and the forces which act 
upon it. We shall suppose generally that the celestial bodies are covered 
with a fluid, and on that hypothesis, which subsists in the case of the earth, 
and which it seems natural to extend to the other bodies of the system of 
the world, we shall determine their figure and the law of gravity at their 
surface. The analysis which we propose to use is a singular application 
of the Calculus of Partial Differences, which by simple differentiation, will 
conduct us to very extensive results, and which with difficulty we should 
obtain by the method of integrations. 

THE ATTRACTIONS OF HOMOGENEOUS SPHEROIDS BOUNDED BY SURFACES 
OF THE SECOND ORDER. 

542. The different bodies of the solar system may be considered as 
formed of shells very nearly spherical, of a density varying according to 
any law whatever ; and we shall show that the action of a spherical shell 
upon a body exterior to it, is the same as if its mass were collected at its 
center. For that purpose we shall establish upon the attractions of sphe 
roids, some general propositions which will be of great use hereafter. 

Let x, y, z be the three coordinates of the point attracted which we 
call ft ; let also d M be the element or molecule of the spheroid, and 
x , y , z the coordinates of this element; if we call o its density, being a 
function of x , y , z independent of x, y, z, we shall have 
d M = | . d x . d y . d z . 

The action of d M upon ft decomposed parallel to the axis of x and 
directed towards their origin, will be 

g d x . d y . d z (x xQ 

KX x ) 2 + (y y ) 2 + (z z ) 2 } 1 
and consequently it will be equal to 

s d x . d y . d z 

d . - J ^-- 



(* x ) 



dx 
calling therefore V the integral 

r- d x . d y . d z 




V (x x) 2 + (y - y ) 2 + (z z ) 2 
extended to the entire mass of the spheroid, we shall have [-. J 



BOOK L] NEWTON S PRINCIPIA. 201 

for the total action of the spheroid upon the point ,, resolved parallel to 
the axis of x and directed towards its origin. 

V is the sum of the elements of the spheroid, divided by their respec 
tive distances from the point attracted ; to get the attraction of the sphe 
roid upon this point, parallel to any straight line, we must consider V as 
a function of three rectangular coordinates, one of which is parallel to this 
straight line, and differentiate this function relatively to this coordinate ; 
the coefficient of this differential taken with a contrary sign, will be the 
expression of the attraction of the spheroid, parallel to the given straight 
line, and directed towards the origin of the coordinate which is parallel to 

it. 

i 
If we represent by ft the function { (x x ) 2 + (y y ) 2 + (z z ) 2 }" 2 ; 

we shall have 

V = //3. f .dx dy dz . 

The integration being only relative to the variables x , y , z , it is evi 
dent that we shall have 



/d^Vx 
(dO 



But we have 
= 



^d x 

v 

in like manner we get 

/d 2 Vx /d 2 Vx /d 2 V 

: \dx 2 / \dy 2 / Uz 2 

This remarkable equation will be of the greatest use in the theory of the fi 
gure of the celestial bodies. We may present it under more commodious 
forms in different circumstances ; conceive, for example, from the origin 
of coordinates we draw to the point attracted a radius which we call g ; 
let d be the angle which this radius makes with the axis of x, and w the 
angle which the plane formed by and this axis makes with the plane of 
x, y; we shall have 

x = P cos. 6 ; v = P sin. 6 cos. -a : z rr P sin. sin. & ; 

* / a 

whence we derive 



z 



s = Vx 2 + y +z 2 ; cos.0= - 7=f ======; tan. * = - 

itivt 

;d 2 
d y 



thus we can obtain the partial differences of g t d, *-, relative to the varia- 

r /d 2 Vx /d 2 VN 
bles x, y, z, and thence get the values of ^j-^r) > \3~y*) 



202 A COMMENTARY ON [SECT. XII. & XIII. 

in partial differences of V relative to the variables P., 6, . Since we shall 
often use these transformations of partial differences, it is useful here to 
lay down the principle of it. Considering V as a function of the variables 
x, y, z, and then of the variables P, 0, , we have 

d PX /d Vv /d 



._ 

- + 



To get the partial differences [r^Yi IT-IJ IT j 5 we must make 

vdx/ VI x/ \dx/ 

x alone vary in the preceding expressions of P, cos. 6, tan. w ; differentiat 
ing therefore these expressions, we shall have 

/d P\ /d &\ sin. /d w\ 

( r~ ) = cos - * ; ( r ) = -- ; ( j - ) = ; 

Vdx/ \dx/ g Vdx/ 

which gives 

/d V\ ,/dV\ sin. tf /d V 

( - ) = cos. 

\dx/ 



Thus we therefore get the partial difference (-=- j , in partial differ- 



ences of the function V, taken relatively to the variables g, 6 t **. Differ 
entiating again this value of f-j J 5 we shall have the partial difference 

j 2 "V7 

( j I )in partial differences of V taken relatively to the variables g, 0, w. 

By the same process the values of (-r F) an( ^ ( . 2 ) ma y be found. 
In this way we shall transform equation (A) into the following one: 

>aVv 

/d 2 Vv cos.*. /dVx Vdw V ^^ gVy /m 

= \m + sinTT U J + inT + e rr * 



And if we make cos. 6 = m, this last equation will become 




dm / 1 

543. Suppose, however, that the spheroid is a spherical shell whose 
origin of coordinates is at the center ; it is evident that V will only de 
pend upon g, and contain neither m nor w t the equation (C) will therefore 
give 



whence by integration we get 



BOOK I.] NEWTON S PRINCIPIA. 203 

A and B being two arbitrary constants. We therefore have 

_(1_Y^ = i. 
\ d / g 2 

expresses, by what precedes, the action of the spherical shell upon 

the point /-, decomposed along the radius g and directed towards the 
center of the shell ; but it is evident that the total action of the shell 

d_V; 

dg 
the total action of the spherical shell upon the point p. 

First suppose this point placed within the shell. If it were at the center 
itself, the action of the shell would be nothing ; we have therefore, 

= 0, or = 0, 

d V 



ought to be directed along this radius ; (. ) expresses therefore 



when = 0, which gives B = 0, and consequently -^ ) = 0, what 

ever may be ; whence it follows that a point placed in the interior of the 
shell, suffers no action, or which comes to the same thing, it is equally at 
tracted on all sides. 

If the point //, is situated without the spherical shell, it is evident, sup 
posing it infinitely distant from the center, that the action of the shell 
upon the point will be the same, as if all the mass of the shell were con 

densed at this center; calling, therefore Mthe mass of the shell, (-, 



or r will become in this case equal to - , which gives B = M ; we have 

S S 

therefore generally relatively to exterior points, 

/d Vx JV1 
-(dg) ? 

that is to say, the shell attracts them as if all its mass were collected at 
its center. 

A sphere being a spherical shell, the radius of whose interior surface ii 
nothing, we see that its attraction, upon a point placed at or above its 
surface, is the same as if its mass were collected at its center. 

This result obtains for globes formed of concentric shells, varying in 
density from the center to the circumference according to any law what 
ever, for it is true for each of the shells : thus since the sun, the planets, 
and satellites may be considered nearly as globes of this nature, they at 
tract exterior bodies very nearly as if their masses were collected into 
their centers of gravity. This is conformable with what has been found by 



204 A COMMENTARY ON [SECT. XII. & XIII. 

observations. Indeed the figure of the celestial bodies departs a lit- 
lle from the sphere, but the difference is very little, and the error which 
results from the preceding supposition is of the same order as" this sup 
position relatively to points near the surface; and relatively to distant 
points, the error is of the same order as the product of this difference by 
the square of the ratio of the radii of the attracting bodies to their 
distances from the points attracted; for we know that the considera 
tion alone of the distance of the points attracted, renders the error of 
the preceding supposition of the same order as tne square of this ratio. 
The celestial bodies, therefore, attract one another very nearly as if their 
masses were collected at their centers of gravity, not only because they 
are very distant from one another relatively to their respective dimensions, 
but also because their figures differ very little from the sphere. 

The property of spheres, by the law of Nature, of attracting as if their 
masses were condensed into their centers, is very remarkable, and we may 
be curious to learn whether it also obtains in other laws of attraction. 
For that purpose we shall observe, that if the law of gravity is such, that 
a homogeneous sphere attracts a point placed without it as if all its mass 
were collected at its center, the same result ought to obtain for a spherical 
shell of a constant thickness; for if we take from a sphere a spherical 
shell of a constant thickness, we form a new sphere of a smaller radius 
with the remainder, but which, like the fonner, shall have the property of 
attracting as if all its mass were collected at its center ; but it is evident, 
that these two spheres can only have this common property, unless it also 
belongs to the spherical shell which forms their difference. The problem, 
therefore, is reduced to determine the laws of attraction according to which 
a spherical shell, of an infinitely small and constant thickness, attracts an 
exterior point as if all its mass were condensed into its center. 

Let be the distance of the point attracted to the center of the spherical 
shell, u the radius of the shell, and d u its thickness. Let d be the angle 
wTiich the radius u makes with the straight line , -a the angle which the 
plane passing through the straight lines f, u, makes with a fixed plane 
passing through , the element of the spherical shell will be u 2 d u . d . 
d 6 sin. 0. If we then call f the distance of this element from the point at 
tracted, we shall have 

f 2 = 2 2 g u cos. 6 + u*. 

Represent by <p (f) the law of attraction to the distance f ; the action of 
the shell s element upon the point attracted, decomposed parallel to g and 
directed towards the center of the shell, will be 



BOOK I.] NEWTON S PRINCIPIA. 205 

, , , . . f u cos. 6 n 
u 2 d u . d -a sin. 6 ~ 7= p (f ) ; 

but we have 

f u cos. 6 _ /d f \ 

f ~ Vd g / 

which gives to the preceding quantity this form 

(0; 



wherefore if we denote fd f <f> (f) by <p, (f) we shall have the whole action 
of the spherical shell upon the point attracted, by means of the integral 
u 2 d ufd -a d 6 sin. d. <p, (f ), differentiated relatively to f, and divided by 
df. 

This integral ought to be taken relatively to w, from = to v equal 
to the circumference, and after this integration it becomes 

2ffu 2 /d 0sin. 6 <p, (f ) ; 
If we differentiate the value of f relatively to d, we shall have 

fdf 

d <J sin. 6 = ---- ; 
S u 

and consequently 

f. p, (f). 



The integral relative to ought to be taken from 6 = to 6 = r, and 
at these two limits we have f = g u, and f = + u ; thus the integral 
relative to f must be taken from f = g utof= + u; let therefore 
/f d f. p, (f) = ^ (f ), we shall have 

2<!f.udu,, .,,. 2 T. u d u 
, (f) = 



The coefficient of d g, in the differential of the second member of this 
equation, taken relatively to g, will give the attraction of the spherical 
shell upon the point attracted ; and it is easy thence to conclude that in 

nature where <f> (f ) = TT this attraction is equal to 
4 it . u 2 d u 

~e~ 

That is to say, that it is the same as if all the mass of the spherical 
shell were collected at its center. This furnishes a new demonstration of 
the property already established of the attraction of spheres. 

Let us determine <p (f ) on the condition that the attraction of the shell 
is the same as if its mass were condensed into its center. This mass 
is equal to 4 T. u 8 d u, and if it were condensed into its center, its action 



206 A COMMENTARY ON [SECT. XII. & XIII. 

upon the point attracted would be 4 T. u * d u . <p (*) ; we shall therefore 
have 




d 
integrating relatively to g, we shall get 

^ (g + u) 4 (g u) = 2 g u/d s . <p (g) + f U, 
U being a function of u and Constants, added to the integral 2 ufd p(g). 
If we represent -^ (? + u) -vj/ (g u) by R, we shall have by differen 
tiating the preceding equation 

d M 



But we have, by the nature of the function R, 
d g R 



du 
wherefore 



or 

d.p(g) _ 1 /d 2 U 



pf , .p(g _ 1 / x 
g d f 2u\du 2 / 

Thus the first member of this equation being independent of u and the 
functions of g, each of its members must be equal to an arbitrary which we 
shall designate by 3 A ; we therefore have 






whence in integrating we derive 

pg = Ag + - g - 

B being a new arbitrary constant. All the laws of attraction in which a 
sphere acts upon an exterior point placed at the distance g from its center, 
as if all the mass were condensed into its center, are therefore comprised 
in the general formula 



it is easy to see in fact that this value satisfies equation (D) whatever may 
be A and B. 

If we suppose A = 0, we shall have the law of nature, and we see that 



BOOK L] NEWTON S PRINCIPIA. 207 

in the infinity of laws which render attraction very small at great dis 
tances, that of nature is the only one in which spheres have the properly 
of acting as if their masses were condensed into their centers. 

O 

This law is also the only one in which a body placed within a spherical 
shell, every where of an equal thickness, is equally attracted on all sides. 
It results from the preceding analysis that the attraction of the spherical 
shell, whose thickness is d u, upon a point placed in its interior, has the 
expression 





To make this function nothing, we must have 

4 (u + f) 4 (u g) = g U, 
U being a function of u independent of g, and it is easy to see that this 

T> 

obtains in the law of nature, where <p (f ) = 5 . But to show that it 

takes place only in this law, we shall denote by (f) the difference of 4> 
(f ) divided by d f, we shall also denote by-vj/ (f) the difference of -vj/ (f) 
divided by d f, and so on ; thus we shall get, by differentiating twice suc 
cessively, the preceding equation relatively to f, 

-V (u + g) 4" (u g) = o. 

This equation obtaining whatever may be u and f, it thence results 
that y (f ) ought to be equal to a constant whatever f may be, and that 
therefore -ty" (f ) = 0. But, by what precedes, 

4/(f) = f.p;(f), 

whence we get 

4/"(f) = 8p(f) +fp (f); 
we therefore have 

= 2p(f) + fp (f); 
which gives by integration 

ic- \ B 
MO = jr> 

and consequently the law of nature. 

554. Let us resume the equation (C) of No. 541. If this equation 
could generally be integrated, we should have an expression of V, which 
would contain two arbitrary functions, which we should determine by 
finding the attraction of a spheroid, upon a point situated so as to facili 
tate this research, and by comparing this attraction with its general ex 
pression. But the integration of the equation (C) is possible only in some 
particular cases, such as that where the attracting spheroid is a sphere, 
which reduces this equation to ordinary differences; it is also possible in 



208 A COMMENTARY ON [SECT. XII. & XIII. 

the case where the attracting body is a cylinder whose base is an oval or 
curve returning into itself, and whose length is infinite. This particular 
case contains the theory of Saturn s ring. 

Fix the origin of g upon the same axis of the cylinder, which we shall 
suppose of an infinite length on each side of the origin. Naming g the 
distance of the point attracted from the axis ; we shall have 

S = I "^ 1 m 2 

It is evident that V only depends on and w, since it is the same for 
all the points relatively to which these two variations are the same ; it 
contains therefore only m inasmuch as g is a function of this variable. 
This gives 

/d V\ __ /d V\ /d P \ %m /d V 

\d m/ " ViTjp vdrn/ " -v/r 

m 2 /d 2 Vx /dV 



the equation (C) hence becomes 



_ _ / 
1 m 2 \d * " (1 _ m t)f VI / r 



whence by integrating we get 

V = <p{ cos. * + | V 1 sin. } + %]// cos. w ^ V 1 sin. } ; 
<f (g ) and >4/ (f 7 ) being arbitrary functions of g , which we can determine 
by seeking the attraction of the cylinder when is nothing and when it 
is a right angle. 

If the base of the cylinder is a circle, V will be evidently a function of 
f independent of v, the preceding equation of partial differences will 
thus become 

M 



which gives by integrating, 

d Vx H 



H being a constant. To determine it, we shall suppose g relatively to 
the radius of the base of the cylinder extremely great, which supposition 
permits us to consider the cylinder as an infinite straight line. Let A be 
this base, and z the distance of any point whatever of the axis of the cy 
linder, to the point where this axis is met by g ; the action of the cylin 
der considered as concentrated or condensed upon its axis, will be, paral 

lei to g , equal to 

/A f . d z 




i 



BOOK I.] NEWTON S PRINCIPIA. 209 

the integral being taken from z = oo to z = co ; this reduces the in 
tegral to - - , ; which is the expression of ( r~7") when g is very con 
siderable. Comparing this with the preceding one we have H = 2 A, 
and we see that whatever is g , the action of the cylinder upon an exterior 

. . 2 A 
point, is j- . 

If the attracted point is within a circular cylindrical shell, of a constant 

thickness, and infinite length, we shall have ( , "\ = ; and since 

\ a g / i 

the attraction is nothing when the point attracted is upon the axis of the 
shell, we have H =.- 0, and consequently, a point placed in the interior of 
the shell is equally attracted on all sides. 

545. We have thus determined the attraction of a sphere and of a 
spherical shell : let us now consider the attraction of spheroids terminated 
by surfaces of the second order. 

Let x, y, z be the three rectangular coordinates of an element of the 
spheroid ; designating d M this element, and taking for unity the density 
of the spheroid which we shall suppose homogeneous, we shall have 
dM = dx.dy.dz. 

Let a, b, c be the rectangular coordinates of the point attracted by the 
spheroid, and denote by A, B, C the attractions of the spheroid upon 
this point resolved parallel to the axes of x, y, z and directed to the origin 
of the coordinates. 

It is easy to show that we have 

A _ rrr (& x) d X . d y . d Z 

{(a x) 2 + (b y) 2 + (c z) 2 }^ 

B =fff (b y) dx. dy. dz 

{(a x) 2 + (b y) 2 + (c z) 2 }* 

C _. rrr (c z) d x . d y . d z 

( a x) 2 + (b y) 2 + (c z) 2 }* 

All these triple integrals ought to be extended to the entire mass of the 
spheroid. The integrations under this form present great difficulties, 
which we can often in part remove by transforming the differentials into 
others more convenient. This is the general principle of such trans 
formations. 

Let us consider the differential function Pdx.dy.dz, P being any 
function whatever of x, y, z. We may suppose x a function of y and z 
and of a new variable p : let p (y, z, p) denote this function ; in this case, 
VOL. II. O 



210 A COMMENTARY ON [SECT. XII. & XIII. 

we shall have, making y and z constant, d x = /3 . d p, j3 being a function 
of y, z and p. The preceding differential will thus become j8 . P . d p . 
d y . d z ; and to integrate it, we must substitute i P, for x, its value 
(y, z, p). 

In like manner we may suppose in this new differential, y = (z, p, q), 
q being a new variable, and (z, p, q) being any function of the three 
variables z, p and q. We shall have, considering z and p constant, 
d y = /3 d q, /3 being a function of z, p, q ; the preceding differential 
will thus take this new form /3 /3 P. d p . d q . d z, and to integrate it, we 
must substitute in j3 P for y its value (z, p, q). 

Lastly we may suppose z equal to 0" (p, q, r), r being a new variable, 
and 0" (p, q, r) being any function whatever of p, q, r. We shall have, 
considering p and q constant, d z = {$" d r, ft" being a function of p, q, r ; 
the preceding differential will thus become /3. /3 . j3". P . d p . d q . d r 
and to integrate it, we must substitute in /3 . fi . P for z its value 0" (p, q, r). 
The proposed differential function is thence transformed to another rela 
tive to the three new variables p, q, r, which are connected with the pre 
ceding by the equations 

x = (y, z, p) ; y = <? (z, p, q) ; z = 0" (p, q, r). 

It only remains to derive from these equations the values of /3, /?, /3". 
For that purpose we shall observe that they give x, y, z, in functions of 
the variables p, q and r ; let us consider therefore the three first variables 
as functions of the three last. Since $" is the coefficient of d r in the dif 
ferential of z, taken by considering p and q constant, we have 

* = ( d df)- 

|S is the coefficient of d q, in the differential of y taken on the supposi 
tion that p and z are constant ; we shall therefore have j6 , by differen 
tiating y on the supposition that p is constant, and by eliminating d r by 
means of the differential of z taken on the supposition that p is constant, 
and equating it to zero. Thus we shall have the two equations 

d y = (TT:) d q + (!r?) d r 



o = 



d 

d z 



/ u Z N j f a z \ j 

(-7- ) d q -f- (-5 ) d r ; 
\dq/ \d T/ 



which give 



d y\ /d z\ /dy\ /d z 
~ ~ ~ 



d y = d q X - * - j 

fPl 

Vdr/ 



BOOK I.] NEWTON S PRINCIPI A. 211 

wherefore 

dyx /dzv /dy 



/yx /zv /y\ 
VdgJ Idr; " " VdrJ 



Finally, /3 is the coefficient of d p, in the differential of x taken on the 
supposition that y and z are constant. This gives the three following 
equations 

d x\ , dx\ , /d 



If we make 

\d p/ \d q/ Vd r/ \d p/ vdr 

/dxx /d yx /d zx /d xx /d y\ /d z 

^d Q/ Vdr/ xdp/ \d q/ xd p/ \d r 



dp 

dx 



^- \dpj vdq/~~VdT; vd q ; \dp 



we shall have 

d p 

d x = s = C. 



wliich gives 

Vlx\ f^ fiz 

Vdq; \dJ "" Mr 
wherefore j8 . jS . $" s and the differential P. d x . d y . d z is transform 
ed into E. P. dp. dq. dr; P being here what P becomes when we 
substitute for x, y, z their values in p, q, r. The whole is therefore re 
duced to finding the variables p, q, r such that the integrations may be 
come possible. 

Let us transform the coordinates x, y, z into the radius drawn from 
the point attracted to the molecule, and into the angles which this ra 
dius makes with given straight lines or with given planes. Let r be 
this radius, p the angle which it forms with a straight line drawn through 
the attracted point parallel to the axis of x, and let q be the angle which 

o -2 



212 A COMMENTARY ON [SECT. XII. & XIII. 

its projection makes on the plane of y, z with the axis of y ; we shall 
have 

x = a r cos. p ; y = b r sin. p cos. q ; z = c r sin. p sin. q. 
We shall then find t = r 2 sin. p, and the differential d x . d y . d z will 
thus be transformed into r 2 sin. p . d p . d q . d r : this is the expres 
sion of the element d M, and since this expression ought to be positive 
in considering sin. p, d p, d q, d r as positive, we must change its sign, 
which amounts to changing that of , and to making e = r 2 sin. p. 

The expressions of A, B, C will thus become 

A =fff<\ r d p d q . sin. p cos. p ; 
B = fff& r dp d q . sin. 2 p cos. p ; 
C = ffj d r dp d q. sin. 2 p sin. q. 

It is easy to arrive by another way at these expressions, by observing 
that the element d M may be supposed equal to a rectangular parallele 
piped, whose dimensions are d r, r d p and r d q sin. p, and by then observing 
that the attraction of the element, parallel to the three axes of x, y, z is 

d M d M dM 

g- cos. p ; r - 2 - sin. p cos. q ; sin. p sin. q. 

The triple integrals of the expressions of A, B, C must extend to the 
entire mass of the spheroid : the integrations relative to r are easy, but 
they are different according as the point attracted is within or without the 
spheroid ; in the first case, the straight line which passing through the 
point attracted, traverses the spheroid, is divided into two parts by this 
point ; and if we call r and r 7 these parts, we shall have 

A =ff(r + r ) d p d q. sin. p cos. p; 
B = ff (r + r ) d p d q . sin. 2 p cos. p ; 
C = ff (r -f- r ) d p d q . sin. 2 p sin. q ; 

the integrals relative to p and q ought to be taken from p and q equal to 
zero, to p and q equal to two right angles. 

In the second case, if we call r, the radius at its entering the spheroid, 
and r the radius at its farther surface, we shall have 

A =ff(v r) d p d q . sin. p cos. p ; 
B ff(^ r) d p d q . sin. 2 p cos. q ; 
C = ff (r r) d p d q . sin. 2 p sin. q. 

The limits of the integrals relative to p and to q, must be fixed at the 
points where r r = 0, that is to say, where the radius r is a tangent 
to the surface of the spheroid. 

546. Let us apply these results to spheroids bounded by surfaces of the 



BOOK I.] NEWTON S PRINCIPIA. 213 

second order. The general equation of these surfaces, referred to the 

three orthogonal coordinates x, y, z is 

OzrA + B.x + C.y + E.z+F. x 2 +H.xy + L.y 2 +M. xz+N. yz+O. z 2 . 

The change of the origin of coordinates introduces three arbitraries, 
since the position of this new origin relating to the first depends upon 
three arbitrary coordinates. The changing the position of the coordi 
nates around their origin introduces three arbitrary angles ; supposing, 
therefore, the coordinates of the origin and position in the preceding 
equation to change at the same time, we shall have a new equation of the 
second degree whose coefficients will be functions of the preceding coeffi 
cients and of the six arbitraries. If we then equate to zero the first 
powers of the coordinates, and their products two and two, we shall de 
termine these arbitraries, and the general equation of the surfaces of the 
second order, will take this very simple form 

x 2 + m y 2 + n z z = k 2 ; 
it is under this form that we shall discuss it. 

In these researches we shall only consider solids terminated by finite 
surfaces, which supposes m and n positive. In this case, the solid is an 
ellipsoid whose three semi-axes are what the variables x, y, z become 

k 

when we suppose two of them equal to zero : we shall thus have k, , , 

V m 

k 
for the three semi-axes respectively parallel to x, to y arid to z. The 

1 1 3 

solid content of the ellipsoid will be 



3 V m n 

If, however, in the preceding equation we substitute for x, y, z their 
values in p, q, r given by the preceding No., we shall have 

r 2 (cos. 2 p + m sin. 1 p cos. 2 q + n sin. 2 p sin. * q) 
2 r (a cos. p + m b sin. p cos. q + n c sin. p sin. q) = k 2 -a 2 -m b s -n c f ; 
so that if we suppose 

I = a cos. p + m b sin. p cos. q -f- n c sin. p sin. q; 
L = cos. * p + m sin. 2 p cos. z q + n sin. 2 p sin. 2 q ; 
R = I 1 + (k 2 a 2 m b 2 n c s ). L 
we shall have 

I + V R 

"tr 

whence we obtain r by taking + , and r by taking ; we shall there 
fore have 

21 , 2 V R 

r + r = T ; r - r = -j . 

O 3 



214 A COMMENTARY ON [SECT. XII. & XIII. 

Hence relatively to the interior points of the spheroid, we get 

. si 
L 



A = 2 f f A P d q I sl " P cos - P - 



TJ _ o r r d P d q . I . sin. 2 p . cos, q 
*fJ ~L~ 

, d p . d q . I . sin. 2 p . sin. q 

~T~ 

and relatively to the exterior points 

A g / / d p . d q . sin, p . cos, p V R 

~~L~ 

p . d q . sin. 2 p cos, q V R 
T 

r 2 /*/* d p d q . sin. 2 p sin. q V R 

yy ~~r~ 

the three last integrals being to be taken between the two limits which 
correspond to R = 0. 

547. The expressions relative to the interior points being the most 
simple, we shall begin with them. First, we shall observe that the semi- 
axis k of the spheroid does not enter the values of I and L ; the values of 
A, B, C are consequently independent ; whence it follows that we may 
augment at pleasure, the shells of the spheroid which are above the point 
attracted, without changing the attraction of the spheroid upon this point, 
provided the values of m and n are constant. Thence results the folloV- 
ing theorem. 

A point placed within an elliptic shell whose interior and exterior sur~ 
faccs are similar and similarly situated, is equally attracted on all sides. 

This theorem is an extension of that which we have demonstrated in 
542, relative to a spherical shell. 

Let us resume the value of A. If we substitute for I and L their va 
lues, it will become 

A /. /.dp.dq.sin.p.cos.p.(acos.p + mbsin.pcos.q + ncsin.psin.q) 

J J cos. 2 p + in sin. 2 p cos. 2 q + n sin. 2 p sin. 2 q 

Since the integrals relative to p and q, must be taken from p and q 
equal to zero, to p and q equal to two right angles, it is clear we have 
generally f P d p . cos. p = 0, P being a rational function of sin. p and 
of cos. z p ; because the value of p being taken at equal distances greater 
and less than the right angle, the corresponding values of P . cos. p are 
equal and have contrary signs ; thus we have 

A = 2 a rr d p.dq.sin. p cos. z p ^ 

J J cos. 2 p + m sin. 2 p cos 2 q -t- n sin " p sin. 2 q * 



BOOK I.] NEWTON S PRINCIPIA. 215 

If we integrate relatively to q from q = to q = two right angles, we 
shall find 



2 a *_ r d p . sin, p cos. * p 

V m n / //_ 1 m \ t , 1 n \ 

/ L/ fi H cos P 1 (l + cos - P ) 

-V \ m r /\ n */ 



an 



integral which must be taken from cos. p rr 1 to cos. p = 1. Let 
cos. p = x, and call M the entire mass of the spheroid ; we shall have 

-, 4r.k 3 , 4 cr 3M , ., . 

by 545, M = - and consequently -= = -j-r ; we shall there 
s/ m n 

fore have 

3aM r 

\. / T==== 




which must be taken from x = 0, to x = 1. 

Integrating in the same manner the expressions of B, C we shall reduce 
them to simple integrals ; but it is easier to get these integrals from the 
preceding expression of A. For that purpose, we shall observe that this 
expression may be considered as a function of a and of the squares k 2 , 

k ~ k 2 

, of the semi-axes of the spheroid, parallel to the coordinates a, b, c 

m n 

of the point attracted ; calling therefore k 2 the square of the semi-axis 
parallel to b, and consequently k 2 . m, and k 2 n the squares of the two 

other semi-axes, B will be a similar function of b, k *, k 2 m, k 2 ; thus 

to get B we must change in the expression of A, a into b, k into k or 

k .1 n . . , 

. , m into . and n into , which gives 
v/ m m m 

m^. x 2 dx 




Let 

t 



x = 



m + (1 m). t 
we shall have 

3bM r t 2 ! dt 



M r _ t 2 ! dt _ 

y- / j 7 5 

/ (i+i^. ,)"(! + !^l. t f 
N m / \ n / 



an integral relative to t which must be taken, like the integral relative to x 

O4- 



216 A COMMENTARY ON [SECT. XII. & XIII. 

from t = to t = 1, because x = gives t = and x = 1, gives t = 1 
Hence it follows that if we suppose 



+ X 2 x 2 ). (1 + x 2 x 2 ) 
we shall have 

__ 3 b M 

~~~~ 



If we change in this expression, b into c, X into X and reciprocally, we 
shall have the value of C. The attractions A, B, C of the spheroid, par 
allel to its three axes are thus given by the following formulas 
_ 3aM ,-, w _ 3 b M /d.xF\ r _ 3 c M 

7 " ^ 5 ( ~ a ~~ 



We may observe that these expressions obtaining for all the interior 
points, and consequently for those infinitely near to the surface, they also 
hold good for the points of the surface. 

The determination of the attractions of a spheroid thus depends only 
on the value of F ; but although this value is only a definite integral, it 
has, however, all the difficulty of indefinite integrals when X and X are 
indeterminate, for if we represent this definite integral, taken from x = 
to x = 1, by <p (X s , X z ), it is easy to see that the indefinite integral will 
be x 3 <p (X x 2 , X 2 x z ), so that the first being given, the second is likewise 
given. The indefinite integral is only possible in itself when one of the 
quantities X, X is nothing, or when they are equal : in these two cases, 
the spheroid is an ellipsoid of revolution, and k will be its semi-axis of 
revolution if X and X are equal. In this last case we have 
^ / x z d x 1 . 

== /i+x x = r^ x - tan "^- 

To get the partial differences fV ), ( ^ ; J, which enter the 
expressions of B, C, we shall observe that 



but when X = X , we have 

/d . x F\ _ /d . x Fx d_x _ d_x- 
V d X ) * \ d x / ; x >/ 

wherefore 



Substituting for F its value, we shall have 
d . X 



BOOK I.] NEWTON S PRINCIPIA. 217 

we shall therefore have relatively to ellipsoids of revolution, whose semi- 
axis of revolution is k, 

A 3a.M., :., 

A = j-^ 3- (X tan. - 1 X) ; 

3 b.M/ 

1-4 I tart > 

~2k 3 . xA ia 
C = 3 c M 

548. Now let us consider the attraction of spheroids upon an exterior 
point. This research presents greater difficulties than the preceding be 
cause of the radical V R which enters the differential expressions, and 
which under this form renders the integrations impossible. We may ren 
der them possible by a suitable transformation of the variables of which 
they are functions ; but instead of that method, let us use the following 
one, founded solely upon the differentiation of functions. 

If we designate by V the sum of all the elements of the spheroid divided 
by their respective distances from the point attracted, and x, y, z the co 
ordinates of the element d M of the spheroid, and a, b, c those of the 
point attracted, we shall have 

V = f JM 

J V (a x) 2 + (b y) 2 + (c z) 2 

Then designating, as above, by A, B, C the attractions of the spheroid 
parallel to the axes of x, y, z, and directed towards their origin, we shall 
have 

A=/ (a x). dM 

{(a. x) 2 + (b y) 2 + ( c 
In like manner we get 

d V, 



whence it follows that if we know V, it will be easy thence to obtain by 
differentiation alone, the attraction of a spheroid parallel to any straight 
line whatever, by considering this straight line as one of the rectangular 
coordinates of the point attracted ; a remark we have already made in 
541. 

The preceding value of V, reduced into a series, becomes 
fi i 2 a x+ 2 b y+ 2 c z x 2 y ! 

J TVT |*+- 2 _L U2 _L * 

v=/ 

" +&C. 



This series is ascending relatively to the dimensions of the spheroid. 



218 A COMMENTARY ON [SECT. XII.&X11I. 

and descending relatively to the coordinates of the point attracted. If we 
only retain the first term, which is sufficient when the attracted point is 
at a very great distance, we shall have 

Y- M 

V a 2 + b 2 + c 2 

M being the entire mass of the spheroid. This expression will be still 
more exact, if we place the origin of coordinates at the center of gravity 
of the sphere ; for by the property of this center we have 

/ x. d M = ; / y. d M = ; / z. d M = ; 

so that if we consider a very small quantity of the first order, the ratio 
of the dimensions of the spheroid to its distance from the point attracted, 
the equation 



V a 2 + b 2 + c 2 
will be exact to quantities nearly of the third order. 

We shall now investigate a rigorous expression of V relatively to ellip 
tic spheroids. 

549. If we adopt the denominations of 544, we shall have 



V =/ =fSS* d r d p d q sin. p = //(r 2 r 2 ) d p d q. sin. p, 
Substituting for r and r their values found in 544, we shall have 
v - rr d p . d q sin, p. I . V R 

: 2 JJ L 2 

Let us resume the values of A B, C relative to the exterior points, and 

given in 546, 

* / d p . d q sin, p cos, p V R 

B = 2 /yy d p . d q sin. g p cos, q V R. 

= 2/7" d P d q sin - 2 P sin - q v R 

Since at the limits of the integrals, we have V R = 0, it is easy to see 
that by taking the first differences of.V, A, B, C relatively to any of the 
six quantities a, b, c, k, m, n, we may dispense with regarding the varia 
tions of the limits ; so that we have, for example, 




for the integral 

/d p sin. p I V R 
LT- 



BOOK I.] NEWTON S PRINCIPIA. 210 

is towards these limits, very nearly proportional to R 2 , which renders 
equal to zero, its differential at these limits. Hence it is easy to see by 
differentiation that if for brevity we make 

aA + bB + cC = F; 

we shall have between the four quantities B, C, F, and V the following 
equation of partial differences, 



We may eliminate from this equation, the quantities B, C, F by means 
of their values 

d Vx /d 



We shall thus get an equation of partial differences in V alone. Let 

therefore 

4*r.k 3 .._ 

V = - == .v = M . v, 
3 V m n 

M being by 545, the mass of the elliptic spheroid ; and for the variables 
m and n let us here introduce 6 and & which shall be such that we have 

1 m i o 1 n i 2 

6= - .k 2 ; 9= - .k 2 ; 
m n 

6 will be the difference of the square of the axis of the spheroid parallel 
to y and the square of the axis parallel to x ; -a will be the difference of 
the square of the axis of z and the square of the axis of x ; so that if we 
take for the axis of x, the smallest of the three axes of the spheroid, V 
and V -a will be its two excentricities. Thus we shall have 



V being considered in the first members of those equations as a function 
of a, b, c, k, m, n ; and v being considered in their second members as a 
function of a, b, c, 6, > k. 



220 A COMMENTARY ON [SECT. XII. & XIII. 

If we make 

n fd v \ , i /d v \ . /d v \ 

Q = a (dli)+ b (db)+ c (d-c) i 

1 T^ 

we shall have F = M Q, and we shall get the values of k(-r , V 

chan s ing in the P recedin g values of k 



-j J , v into Q. Moreover V and F are homogeneous functions in 

a, b, c, k, V d, V -a of the second dimension, for V being the sum of the 
elements of the spheroid, divided by their distances from the point at 
tracted, and each element being of three dimensions, V is necessarily of 
two dimensions, as also F which has the same number of dimensions as 
V ; v and Q are therefore homogeneous functions of the same quantities 
and of the dimension 1 ; thus we shall have by the nature of homo 
geneous functions, 



an equation which may be put under this form 



We shall have in like manner 



then, if in equation (1) we substitute for V, F and their partial differences; 

k 2 k 2 

if moreover we substitute . , . . for m and , 9 - for n, we shall have 

k 2 + d k 2 + w 



550. Conceive the function v expanded into a series ascending rela 
tively to the dimensions k, V 6, V -a of the spheroid, and consequently 
descending relatively to the quantities a, b, c : this series will be of the 



following form : 



v = U <> + U (1 + U + U ^ + &c. ; 

U (0) , U (1) , &c. being homogeneous functions of a, b, c, k, V 6, V &, and 
separately homogeneous relatively to the three first and to the three last 



BOOK I.] 



NEWTON S PRINCIPIA. 



221 



of these six quantities; the dimensions relative to the three first always 
decreasing, and the dimensions relative to the three last increasing con 
tinually. These functions being of the same dimension as v, are all of the 
dimension 1. 

If we substitute in equation (2) for v its preceding expanded value ; if 
we call s the dimension of U (i) in k, V t), V &, and consequently s 1 
its dimension in a, b, c ; if in like manner we name s the dimension of 
|J( + i) i n k ? V 6, V **, and consequently s 1 its dimension in a, b, 
c ; if we then consider that by the nature of homogeneous functions we 
have 




we shall have, by rejecting the terms of a dimension superior in k, V 0, 
V ar to that of the terms which we retain, 



U a + 1) = _k 



(3) 



s . 



This equation gives the value of U (i + 1 \ by means of U (i) and of its 
partial differences ; but we have 



(a 2 + b 2 + C 2 )2 
since, retaining only the first term of the series, we have found in 548, that 

v = M .. 

(a 2 + b 2 + c 2 ) 2 

Substituting therefore this value of U (0) in the preceding formula, we 
shall get that of U (1 > ; by means of that of U (1) we shall have that of U (2) 
and so on. But it is remarkable that none of these quantities contains k: 
for it is evident by the formula (3) that U (0) , not containing U (n , does 
not contain it ; that U (1) not containing it, U (2) will not contain it, and so 
on ; so that the entire series U (0) + U (1) + &c. is independent of k, or 

which is the same thing ( = 0. The values of v, 



222 A COMMENTARY ON [SECT. XII. & XIII. 

(cTc) aie t ^ erefore the same f r all elliptic spheroids similarly si 
tuated, and which have the same excentricities V 0, V & ; but M H-^ 

^d a/ 

AT /d v\ ,. . /d v\ 
- M Vd~j~) M ^ j-^J > express by 548,the attractions of the spheroid 

parallel to its three axes; therefore the attractions of different elliptic 
spheroids which have the same center, the same position of the axes and 
the same excentricities, upon an exterior point, are to one another as their 
masses. 

It is easy to see by formula (3) that the dimensions of U <>, U C1 >, U C2 >, 
&c. in V6 and V *, increase two units at a time, so that s = 2 i, s = 2 i 
moreover we have by the nature of homogeneous functions 



this formula will therefore become 




By means of this equation, we shall have the value of v in a series very 
convergent, whenever the excentricities V d, V & are very small, or when 
the distance Va 2 + b z + c 2 of the point attracted from the center of 
the spheroid is very great relatively to the dimensions of the spheroid. 

If the spheroid is a sphere, we shall have = 0, and = 0, which 
give U (1) = 0, U (2) = 0, &c. ; wherefore 

V = U W = 1 ; 

V a 2 + b 2 + c a 
and 

M 



V = 



V a 2 + b 2 + c 2 

whence it follows that the value of V is the same as if all the mass of the 
sphere were condensed into its center, and that thus, a sphere attracts any 
exterior point, as if all its whole mass were condensed into its center ; a 
result already obtained in 542. 

551. The property of the function of v being independent of k, fur 
nishes the means of reducing its value to the most simple form of which it 
is susceptible ; for since we can make k vary at pleasure without changing 
this value, provided the spheroid retain the same excentricities, V 6 and 



BOOK I.] NEWTON S PEIINCIPIA. 223 

V *r, we may suppose k such that the spheroid shall be infinitely flatten 
ed, or so contrived that its surface pass through the point attracted. In 
these two cases, the research of the attractions of the spheroid is rendered 
more simple; but since we have already determined the attractions of elliptic 
spheroids, upon points at the surface, we shall now suppose k such that 
the surface of the spheroid passes through the point of attraction. 

If we call k , m , n relatively to this new spheroid what in 545, we 
named k, m, n relatively to the spheroid we there considered ; the condi 
tion that the point attracted is at the surface, and that also a, b, c are the 
coordinates of a point of the surface, will give 

a* + m b z + n c 2 = k 2 ; 

and since we suppose the excentricities V 6 and V w to remain the same, 
we shall have 

whence we obtain 

k 2 , k 2 

YYV ^ tl """ " 
HI 1/0 I > ll 1/0 , 



we shall therefore have to determine k , the equation 



It is easy hence to conclude that there is only one spheroid whose sur 
face passes through the point attracted, 6 and -a remaining the same. For 
if we suppose, which we always may do, that 6 and are positive, it is 
clear that augmenting in the preceding equation, k 2 by any quantity which 
we may consider an aliquot part of k /2 , each of the terms of the first 
member of this equation, will increase in a less ratio than k 2 ; therefore 
if in the first state of k 2 , there subsist an equality between the two mem 
bers of this equation, this equality will no longer obtain in the second 
state ; whence it follows that k 2 is only susceptible of one real and posi 
tive value. 

Let M be the mass of the new spheroid, and A , B 7 , C its attractions 
parallel to the axes of a, b, c ; if we make 
1 _ m 1 n 

_ _ \ 2 . . ^ * 

m n 

~ J V(l + x 2 . x 2 ). (1 + x". x *) ; 
by 547, we shall have 

_ 3 a M F B , = 3b JV 



221 A COMMENTARY ON [SECT. XII. & Xlll. 

Changing in these values of A , B , C , M into M, we shall have by 
the preceding No., the values of A, B, C relatively to the first spheroid 
but the equations 

1 m 1 n 



m n 

give 



, /2 
k 



> 5 _ * - / 2 __ W 

= p- 8 ; - k / 2 ; 

k 2 being given by equation (5) which we may put under this form 

we shall therefore have 

3 a M */- 3b M/d.xF^ 3_cM 1 " ~ 

1/3 Jc J -D - i . I r J 5 ^ i / j 



These values obtain relatively to all points exterior to the spheroid, and 
to extend them to those of the surface, and even to the interior points 
we have only to change k to k. 

If the spheroid is one of revolution, so that 6 = w, the formula (5) 
will give 



2 k /2 = a 2 +b 2 + c 2 6 + V(a 2 + b 2 +c 2 6) 2 + 4 a 2 . 
and by 547. we shall have 

3 a M /, a 

~ k ~\T s ( * 

3 b M >. 





r 



3 c M 



Thus we have terminated the complete theory of the attractions of el 
liptic spheroids ; for all that remains to be done is the integration of the 
differential expression of F, and this integration in the general sense is 
impossible, not only by known methods, but also in itself. The value of F 
cannot be expressed in finite terms by algebraic, logarithmic or circular 
quantities ; or which it tantamount, by any algebraic function of quantities 
whose exponents are constant, nothing or variable. Functions of this kind 
being the only ones which can be expressed independently of the symbol 
J] all the integrals which cannot be reduced to such functions, are impos 
sible in finite terms. 

If the elliptic spheroid is not homogeneous, and if it is composed of 
elliptic shells varying in position, excenlricity and density according to 
any law whatever, we shall have the attraction of one of its shells, by de- 



BOOK I.] NEWTON S PRINCIPIA. 225 

termining as above the difference of the attractions of two homogeneous 
elliptic spheroids, having the same density as the shell, one of which shall 
have for its surface the exterior surface of the shell, and the other the in 
terior surface of the shell. Then summing this differential attraction, we 
shall have the attraction of the whole spheroid. 

THE DEVELOPEMENT INTO SERIES, OF THE ATTRACTIONS OF ANY 
SPHEROIDS WHATEVER. 

552. Let us consider generally the attractions of any spheroids what 
ever. We have seen in No. 547, that the expression V of the sum of the 
elements of the spheroid, divided by their distances from the attracted 
points, possesses the advantage of giving by its differentiation, the attrac 
tion of this spheroid parallel to any straight line whatever. We shall see 
moreover, when treating of the figure of the planets, that the attraction of 
their elements presents itself under this form in the equation of their equi 
librium ; thus we proceed particularly to investigate V. 

Let us resume the equation of No. 548, 

v - r dM 

J V (a x) 2 + (b y) 2 + (c z) 

a, b, c being the coordinates of the point attracted; x, y, z those of the 
element d M of the spheroid ; the origin of coordinates being in the in 
terior of the spheroid. This integral must be taken relatively to the va 
riables x, y, z, and its limits are independent of a, b } c; hence we shall 
find by differentiation, 



an equation already obtained in 541, 

Let us transform the coordinates to others more commodious. For 
that purpose, let r be the distance of the point attracted from the origin 
of coordinates ; the angle which the radius r makes with the axis of a ; 
<* the angle which the plane formed by the radius and this axis, makes 
with the plane of the axis of a, and of b ; we shall have 

a = r cos. 6 ; b = r sin. 6 cos. 6 ; c = r sin. 6 sin. -a. 

If in like manner we name R, tf, -a what r, d, -a become relatively to 
the element d M of the spheroid ; we shall have 

x = R cos. ff ; y = R sin. & cos. -a ; z = R sin. 6 . sin. -a . 

Moreover, the element d M of the spheroid is equal to a rectangular 

parallelepiped whose dimensions are d R, R d , R d -a sin. 6 , and con- 
VOL. II. P 



226 A COMMENTARY ON [SECT. XII. & XIII. 

sequently it is equal to g. R 2 . d R. d tf. d . sin. , g being its density; we 
shall thus have 

V - fff g R *. d R . d (f. d sin. tf_ _ __ 

JJJ ^ r a g r R cos> (?. cos. +sin. sin. f cos. (t/ )J + R 2 

the integral relative to R must be taken from R = to the value of R at 
the surface of the spheroid ; the integral relative to */ must be taken from 
a = to a equal to the circumference ; and the integral relative to V 
must be taken from 6 = to (f equal to the semi-circumference. Differ 
entiating this expression of V, we shall find 



- f!\ , cos^ ,d Vx \d-* /d 2 . r \ 

~ \d t*) + sin. & \d~JJ "" sin. 2 <J + r ( d r 2 
an equation which is only equation (1) in another form. 
If we make cos. 6 = m, we may give it this form 




* ( 

We have already arrived at these several equations in 541. 
553. First, let us suppose the point attracted to be exterior to the sphe 
roid. If we wish to expand V into a series, it ought in this case, to de 
scend relatively to powers of r, arid consequently to be of this form 

u*> 



Substituting this value of V in equation (3) of the preceding No., the 
comparison of the same powers of r will give, whatever i may be 




It is evident from the integral expression alone of V that U (i) is a ra 
tional and entire function of m, V 1 m 2 . sin. -or, and V 1 m~ 2 . cos. w, 
depending upon the nature of the spheroid. When i = 0, this function 
becomes a constant ; and in the case of i = 1, it assumes the form 

H m + H V 1 m 2 . sin. * + H" V 1 m 2 . cos. w ; 
H, H , H ff being constants. 

To determine generally U call T the radical 

^^^ __ 1 _ 
Vr* 2 R r ic 



BOOK I.] NEWTON S PRINCIP1A. 227 

we shall have 




dm 

This equation will still subsist if we change 6 into 6 , * into * , and re 
ciprocally ; because T is a similar function of 1 , & and of 0, -a. 

If we expand T, in a series descending relatively to r, we shall have 

TJ 



Q W being, whatever i may be, subject to the condition that 




= _ 

dm / m 

and moreover it is evident, that Q (i) is a rational and entire function of m, 
and V 1 m 2 . cos. (& ) : Q (i) being known, we shall have U (l) by 
means of the equation 

U =fg R (i + 2) . d R . d . d ff . sin. 6 . Q . 

Now suppose the point attracted in the interior of the spheroid : we 
must then develope the integral expression of V, in a series ascending re 
latively to r, which gives for V a series of the form 

V = v (0 > + r . v (1 > + r 2 . v (2 ) + r 3 . v C3 > + &c. 

v (l) being a rational and whole function of m, V I m 2 . sin. and 
VI m z cos. -, which satisfies the same equation of partial differences 
that U (i) does ; so that we have 




dm / 1 m 

To determine v (i) , we shall expand the radical T into a series ascending 
according to r, and we shall have 

O W r r ~ 

T = ^ + Q . ^ 2 + Q (2) - ^3 + &c- 

the quantities Q (0) , Q U) , Q (2) , &c. being the same as above ; we shall 
therefore get 

/g.d R.dw . dO .sin. 
~ 



~ 



But since the preceding expression of T is only convergent so long as 
R is equal to or greater than r, the preceding value of V only relates to the 
shells of the spheroid, which envelope the point attracted. This point 
being exterior, relatively to the other shells, we shall determine that part 

of V which is relative to them by the first series of V. 

P2 



228 A COMMENTARY ON [SECT. XII. & XIII. 

554. First let us consider those spheroids which differ but very little 
from the sphere, and determine the functions U (0 -, U (1) , U (2) , Sac. v (), 
v (1 >, v (2 >, &c. relatively to these spheroids. There exists a differential 
equation in V, which holds good at their surface, and which is remarkable 
because it gives the means of determining those functions without any in 
tegration. 

Let us suppose generally, that gravity is proportional to a power n of 
the distance ; let d M be an element of the spheroid, and f its distance 
from the point attracted; call V the integraiyf n + 1 d M, which shall ex 
tend to the entire mass of the spheroid. In nature we have n = 2, 

/d M 

it becomes J p , and we have expressed it in like manner by V in the 

preceding Nos. The function V possesses the advantage of giving, by its 
differentiation, the attraction of the spheroid, parallel to any straight line 
whatever ; lor considering f as a function of the three coordinates of the 
point attracted perpendicular to one another, and one of which is parallel 
to this straight line. Call r this coordinate, the attraction of the spheroid 

1 f 

along r and directed towards its origin, will bey. f n . f-f } d M. Con 
sequently it will be equal to (-, j , which, in the case of nature, 

becomes ( ) , conformably with what has been already shown. 

Suppose, however, that the spheroid differs very little from a sphere of 
the radius a, whose center is upon the radius r perpendicular to the sur 
face of the spheroid, the origin of the radius being supposed to be arbi 
trary, but very near to the center of gravity of the spheroid; suppose, 
moreover, that the sphere touches the spheroid, and that the point at 
tracted is at the point of contact of the two surfaces. The spheroid is 
equal to the sphere plus the excess of the spheroid above the sphere ; but 
we may conceive this excess as being formed of an infinite number of 
molecules spread over the surface of the sphere, these molecules being 
supposed negative wherever the sphere exceeds the spheroid; we shall 
therefore have the value of V by determining this value, 1st, relatively to 
the sphere ; 2dly, relatively to the different molecules. 

Relatively to the sphere, V is a function of a, which we denote by A ; 
if we name d m one of the molecules of the excess of the spheroid above 
the sphere, and f its distance from the point attracted ; the value of V rela- 



BOOK L] NEWTON S PRINCIPIA. 229 

tive to this excess will be/. f n + l . d m ; we shall therefore have, for the 
entire value of V, relative to the spheroid, 
V = A+/. fn + i.dm. 

Conceive that the point attracted is elevated by an infinitely small 
quantity d r, above the surface of the spheroid and the sphere upon r or a 
produced ; the value of V, relative to this new position of the attracted 
point, will become 



A will increase by a quantity proportional to d r, and which we shall re 
present by A . d r. Moreover, if we name 7 the angle formed by the two 
radii drawn from the center of the sphere to the point attracted, and to 
the molecule d m, the distance f of this element or molecule from the point 
attracted, will be in the first position of the point, equal to 

V 2 a 2 (1 cos. 7) ; 
in the second position it will be 

V (a + d r) 2 2 a (a -f- d r) cos. 7 + a 2 , 
or 



the integral/, f n + 1 d m, will thus become 

{ + ^- 

we shall therefore have 



substituting for/, f n + ! . d m, its value V A. we shall have 

f (n + 1} A n+ 1 



In the case of nature, the equation (1) becomes 



The value of V relative to the- sphere of radius a, is, by 550, equal to 
-~- , which gives A = ^ a ; A = ^~ ; we shall therefore 
get 



* 

We must here observe that this equation obtains, whatever may be the 
position of the straight line r, and even in the case where it is not perpen- 

r 3 



230 A COMMENTARY ON [SECT. XII. & XIII. 

dicular to the surface of the spheroid, provided that it passes very near its 
center of gravity, for it is easy to see that the attraction of the spheroid, 
resolved parallel to these straight lines, and which, as we have seen, is 

equal to (~TT) > * s > whatever may be their position, always the same, to 

quantities nearly of the order of the square of the excentricity of the 
spheroid. 

555- Let us resume the general expression of V of 553, relative to a 
point attracted exterior to the spheroid, 

U& , U< . U< 8 > , 
V = + -77- + "73 - + &c. 

the function U (i) being, whatever i may be, subject to the equation of par 
tial differences 





dm / 1 m 2 

By differentiating the value of V relatively to r, we have 

/d Vx TJ(> , 2U (1 > , 3 U . 

(-i ) = T M 4 r- &c. 

v d r / r 2 r 3 r 4 

Let us represent by a (1 + ay) the radius drawn from the origin of 
r to the surface of the spheroid, being a very small constant coefficient, 
whose square and higher powers we shall neglect, and y being a function 
of m and depending on the nature of the spheroid. We shall have to 

4 1 IT 3. 

quantities nearly of the order , V = ; whence it follows that in the 

A 3 

preceding expression of V, 1st, the quantity U (0) is equal to plus a very 

small quantity of the order , and which we shall -denote by U W) ; 
2dly, that the quantities U Cl) , U (2) , &c. are small quantities of the order a. 
Substituting a (1 + a y) for r in the preceding expressions of V and of 

> f-r V and neglecting quantities of the order a 2 , we shall have rela 
tively to an attracted point placed at the surface 

i T-r /* a / \ > ^^ i ^^ r ^^ O 



If we substitute these values in equation (2) of the preceding No. we 
shall have 

2 , - U/(0) 3 U U) 5 U ( * ] + I*?. ^^ & C 

fft ^~~a a 2 a 3 a 4 



BOOK I.] NEWTON S PRINCIPIA. 231 

It thence follows that the function y is of this form 

y = Y<> + YW + Y< 8 > + &c. 

the quantities Y (0) , Y W, Y (2 \ &c. as well as U (0 >, U (1 >, &c. being subject 
to the equation of partial differences 




m 

this expression of y is not therefore arbitrary, but it is derived from the 
developement of the attractions of spheroids. We shall see in the follow 
ing No. that y cannot be thus developed except in one manner only ; we 
shall therefore have generally, by comparing similar functions, 

(i) _ 4 av . + 3 Y p) . 
2 i + 1 

whence, whatever r may be, we derive 



To get V, therefore, it remains only to reduce y to the form above de 
scribed ; for which object we shall give, in what follows, a very simple 
method. 

If we had y = Y (i) , the part of V relative to the excess of the spheroid 
above the sphere whose radius is a, or which is the same thing, relative to 
a spherical shell whose radius is a, and thickness a a y, would be 

-TO~-~I j\ i + i 5 this value would consequently be proportional to y, 

and it is evident that it is only in this case that the proportionality can 
subsist. 

556. We may simplify the expression Y (0) + Y (1 > + Y + &c. of y, 
and cause to disappear the two first terms, by taking for a, the radius of a 
sphere equal in solidity to the spheroid, and by fixing the arbitrary origin 
of r at the center of gravity of the spheroid. To show this, we shall ob 
serve that the mass M of the spheroid supposed homogeneous, and of a 
density represented by unity, is by 552, equal to/R 2 d R d m d w, or to 
^./R 3 d m d , R being the radius R produced to the surface of the 
spheroid. Substituting for R its value a (1 + a y) we shall have 

M = i- 3 aa 3 dmd*r. 



All that remains to be done, therefore, is to substitute for y its value 
Y (0) + Y (1) + &c. and then to make the integrations. For this purpose 
here is a general theorem, highly useful also in this analysis. 

r 1 



232 A COMMENTARY ON [SECT. XII. & XIII. 

" If Y (i) and Z w be rational and entire functions of m, V 1 m 2 . sin. -a 
" and V 1 m 2 . cos. *r, which satisfy the following equations : 




= _ 

dm 
" we shall have generally 

/Y (1) . Z >.dmd*r = 0,~ 

" whilst i and i are whole positive numbers differing from one another. 
" the integrals being taken from m = 1 to m = 1, and from = 
" to = 2 ." 

To demonstrate this theorem, we shall observe that in virtue of the first 
of the two preceding equations of partial differences, we have 



/Y . 




a 

m . d 



_m 
But integrating by parts relatively to m we have 




and it is clear that if we take the integral from m = 1 to m = 1, the 
second member of this equation will be reduced to its last term. In like 
manner, integrating by parts relatively to w, we get 



and this second member also reduces to its last term, when the integral 



BOOK I.] NEWTON S PRINCIPIA. 233 

/d Y (i \ 
is taken from w =r to r = 2 *, because the values of Y (l) , ( 1 - }, 

N Cl -at / 

Z ( % ( , \ are the same at these two limits; thus we shall have 
/Y. Z^.dm. d = 





dm 

whence we derive, in virtue of the second of the two preceding equations 
of partial differences, 

/ Y . Z O 1 ). d m . d w = 1 T jjll ) - ./ Y W. Z M. d m . d * , 

we therefore have 

=/Y. Z dm. d *, 
when i is different from i . 

. Hence it is easy to conclude that y can be developed into a series of 
the form Y (0 > + Y (1 > + Y + &c. in one way only; for we have 
generally 

fy . Z d m d = / Y . Z d m . d ; 

If we could develope y into another series of the same form, Y/ 0) + 
Y / U) + Y 7 + &c. we should have 

/y.Z> =/,. Zdm.d^ ; 
wherefore 

/Y, W. Z ). d m d tr rr /Y W. Z ) d m . d tr. 

But it is easy to perceive that if we take for Z (l) the most general 
function of its kind, the preceding equation can only subsist in the case 
wherein Y, (i > = Y (i) ; the function y can therefore be developed thus in 
only one manner. 

If in the integraiy y d m . d w, we substitute for y its value Y (0) + Y (1) 
-f Y + &c., we shall have generally f Y (i) d m . d , i being 
equal to or greater than unity ; for the unity which multiplies d m . d 
is comprised in the form Z ^ D , which extends to every constant and quan 
tity independent of m and *. The integraiy y d m . d * reduces there 
fore toy Y (0) d m . d w, and consequently to 4 T Y (0) ; we have there 
fore 

M = f era 3 + 4 air a 3 . Y ^ ; 

thus, by taking for a, the radius of the sphere equal^in solidity to the sphe 
roid, we shall have Y (0) = 0, and the term Y (0) will disappear from the 
expression of y. 



234 A COMMENTARY ON [SECT. XII. & XIII. 

The distance of the element d M, or R 2 . d R d m . d w, from the 
plane of the meridian from whence we measure the angle w, is equal to 
R V 1 m 2 . sin. *; the distance of the center of gravity of the sphe 
roid from this plane, will be therefore/ R 3 d R d m . d VI m 2 . sin. *r, 
and integrating relatively to R, it will be ^/R 4 d m . d -a VI m 2 sin. *, 
R being the radius R produced to the surface of the spheroid. In like 
manner the distance of the element d M from the plane of the meridian 
perpendicular to the preceding, being R V 1 m 2 . cos. *, the distance 
of the center of gravity of the spheroid from this plane will be \ f R /4 
clm.dw. V I m 2 . cos. *. Finally, the distance of the element d M 
from the plane of the equator being m, the distance of the center of gra 
vity of the spheroid from this plane will be \f R 4 m . d m . d . These 
functions m, V I m 2 . sin. , V 1 m 2 . cos. w, are of the form Z (I >, 
Z (1) being subject to the equation of partial differences 




J ] 



+ 2Z 



dm / \ m 

If we conceive R 4 developed into the series N (0) + N (1) + N + &c. 
N (i) being a rational and entire function of m, VI m" 2 . sin. ?r, 
V 1 m 2 . cos. "vr, subject to the equation of partial differences. 

d 




dm y 1 m 

the distances of the center of gravity of the spheroid, from the three 
preceding planes, will be, in virtue of the general theorem above demon 
strated, 

i/N<. dm. d. V 1 m 2 . sin. *r, 

4/N (1) . d m . d *> . V 1 m 2 . cos. * ; 
. d m. d . 



N C1) is, by No. 553, of the form A m + B VI m 2 . sin. -a -f 
C V 1 m 2 . cos. w, A, B, C being constants ; the preceding distances 

will thus become -^ . B, -^ . C, -^- . A. The position of the center of 
o o o 

gravity of the spheroid, thus depends only on the function N C1) . This 
gives a very simple way of determining it. If the origin of the radius R 
is at the center; this origin being upon the three preceding planes, the 
distances of the center of gravity from these planes will be nothing. This 
gives A = 0, B = 0, C = 0; therefore N (1) = 0. 



BOOK I.] NEWTON S PRINCIPIA 235 

These results obtain whatever may be the spheroid : when it is very 
little different from a sphere, we have R = a (1 + y), and R 4 = 
a 4 (1 + 4 a y) ; thus, y being equal to Y (0) + Y (1 > + Y + &c., we 
have N C1) 4 a a 4 Y (l \ the function Y (1 disappears, therefore, from the 
expression of y, when we fix the origin of R at the center of gravity of 
the spheroid. 

557. Now let the point attracted be in the interior of the spheroid, we 
shall have by 553 

V = v <> + r . v (1 > + r 2 . v & + r 3 v (3 > -f &c. 

r d R . d J . d (f . sin. tf . Q ^ 
v uj . j i _ l . 

Suppose that this value of V is relative to ashell whose interior surface is 
spherical and of the radius a, and the radius of whose exterior surface is 
a (1 y); the thickness of the shell is a a y. If we denote by y what 
y becomes when we change Q, -a into <) , & , we may, neglecting quantities 
of the order a 2 , change r into a, and d R into a a y , in the integral ex 
pression of v W ; thus we shall have 

v W = -j^/y d w . d (f . sin. (f . Q . 

a 

Relatively to a point placed without the spheroid, we have, by 553, 

v - u(0) H!l! 

~~r~ + T~ ~*~ C } 

U (i) =fR l + 2 . dR.d~r.dff. sin. 8. Q (i >. 

If we suppose this value of V relative to a shell, whose interior and ex 
terior radii are respectively a, a (1 + a y), we shall have 

U (i1 = . a + ./y. d */. d 6 f . sin. V. Q >; 
wherefore 

U-W 

y W 

We have by 555 



U w - - 
therefore 


i a w A .1 


2i + 1 

4 a 9 Y W 



(2i+ iFaT 
which gives 

( r r 2 

I + 3~a "^"Sa" 2 

To this value of V we must add that which is relative to the spherical 

shell of the thickness a r which envelopes the attracted point plus that 

which is relative to the sphere of radius r, and which is below the same 



236 A COMMENTARY ON [SECT. XII. & XIII. 

point If we make cos. tf = m , we shall have, relatively to the first of 
the two parts of V, 

r d R . d */ . d m . Q w 

v * ~ / - _ 

-J R i-.l 

an integral which, relative to m , must be taken from m = 1 to m == 1 
Integrating relative to R, from R = r to R = a, we shall have 



m. ; 

But we have generally, by the theorem of the preceding No., 
yd & . d m . Q (i) = when i is equal to or greater than unity; when 
i = 0, we have, by 553, Q (n = 1 ; moreover the integration relative to 
/ must be taken from -of = to & = 2 <K ; we shall therefore have 

v<> = 2 * (a 2 r 2 ). 

This value of v (0) is that part of V which is relative to the spherical shell 
whose thickness is a r. 

The part of V which is relative to the sphere whose radius is r is equal 
to the mass of this sphere, divided by the distance of the attracted point from 

4 .. 2 

its center : it is consequently equal to - . Collecting the different 

9 

parts of V,we shall have its whole value 

. (4) 



Suppose the point attracted, placed within a shell very nearly spherical, 
whose interior radius is 

a + a a fY + Y> + Y + &c.} 
and whose exterior radius is 

a + a [Y W + Y W + Y + &c.} 

The quantities a a Y {0) and a af Y (0) may be comprised in the quanti 
ties a, of. Moreover, by fixing the origin of coordinates at the center of 
gravity of the spheroid whose radius is 

a+ a fY<> + Y> + &c.$, 

we may cause Y (I) to disappear from the expression of this radius ; and 
then the interior radius of the shell will be of this form, 

a + aa {Y + Y + &c.}, 
and the exterior radius will be of the form, 

a + a Y /(1 > -f Y + &c.}. 

We shall have the value of V relative to this shell, by taking the differ 
ence of the values of V relative to two spheroids, the smaller of which 
shall have for the radius of its surface the first quantity, and the greater 



BOOK I.] NEWTON S PR1NCIPIA. 237 

the second quantity for the radius of its surface ; calling therefore A . V, 
what V becomes relatively to this shell, we shall have 



If we wish that the point placed in the interior of the shell, should be 
equally attracted on all sides, A . V must be reduced to a constant inde 
pendent of r, 6, zr ; for we have seen that the partial differences of A . V, 
taken relatively to these variables, express the partial attractions of the 
shell upon the point attracted ; we therefore, in this case have Y (1) = 0, 
and generally 

Y W = f--V-2. y 0). 
> a / 

so that the radius of the interior surface being given, that of the exterior 
surface will be found. 

When the interior surface is elliptic, we have Y (3) = 0, Y (4) = 0, &c. 
and consequently Y /(3) = 0, Y /(4) = 0; the radii of the two surfaces, in 
terior and exterior, are therefore 

aU + Y<*}; a {l + Y>J; 

thus we see that these two surfaces are similar and similarly situated, 
which agrees with what we found in 547. 

558. The formulas (3), (4) of Nos. 555, and 557, comprehend all the 
theory of the attractions of homogeneous spheroids, differing but little from 
the sphere; whence it is easy to obtain that of heterogeneous spheroids, 
whatever may be the law of the variation of the figure and density of their 
shells. For that purpose let a ( 1 + a y) be the radius of one of the shells 
of a heterogeneous spheroid, and suppose y to be of this form 

Y<> + Y 1 + Y< 2 > -f- &c. 

the coefficients which enter the quantities Y (0) , Y (1) , &c. being functions 
of a, and consequently variable from one shell to another. If we differ 
entiate relatively to a, the value of V given by the form (3) of No. 555 ; 
and call g the density of the shell whose radius is a (1 + y), being a 
function of a only ; the value of V corresponding to this shell will be, for 
an exterior attracted point, 



this value will be, therefore, relatively to the whole spheroid, 

.; . . (5) 



the integrals being taken from a = to that value of a which subsists at 
the surface of the spheroid, and which we denote by a. 



238 A COMMENTARY ON [SECT. XII. & XliL 

To get the part of V relative to an attracted point in the interiorwf the 
spheroid, we shall determine first the part of this value relative to all the 
shells to which this point is exterior. This first part is given by formula 
(5) by taking the integral from, a = to a = a, a being relative to the 
shell in which is the point attracted. We shall find the second part of V 
relative to all the shells in the interior of which is placed the point attract 
ed, by differentiating the formula (4) of the preceding No. relatively to a; 
then multiplying this differential by , and taking the integral from a = a, 
to a = a, the sum of the two parts of V will be its entire value relative to 
an interior point, which sum will be 



~ Y> + &c.. (G) 

the two first integrals being taken from a = to a = a, and the two last 
being taken from a = a to a = a; after the integrations, moreover, we 

must substitute a for r in the terms multiplied by , and - * for 

in the term - f P d . a 3 . 
r 3 r j 

559. Now let us consider any spheroids whatever. The research of 
their attraction is reduced, by 553, to forming the quantities U (i) and v ^ , 
by that No. we have 

U r=/gR i + 2 . d Rdm dt* . Q; 

in which the integrals must be taken from R = to its value at the sur 
face, from m = 1 to m = 1, and from & to */ = 2 it. 

To determine this integral, Q W must be known. This quantity may 
be developed into a finite function of cosines of the angle & /, and of 
its multiples. Let /3 cos. n (r ) be the term of Q W depending on 
cos. n (a /), being a function m, m . If we substitute for Q (i) its 
value in the equation of partial differences in Q (i) of No. 553, we shall 
have, by comparing the terms multiplied by cos. n (& ), this equation 
of ordinary differences, 



R w 
Q (i) being the coefficient of - . + t , in the developement of the radical 



1 



V r " 2 Rr\m m + V 1 m 2 . V~l m 2 . cos. (* * } + R ! 



BOOK I.J NEWTON S PRINCIPIA. 239 

The term depending on cos. n (& */), in the developement of this 
radical, can only result from the powers of cos. (& & ), equal to n, n-f-2, 
n + 4, &c. ; thus cos. (a /) having the factor V I m 2 , /3 must have 

the factor (1 m 2 ) ^. It is easy to see, by the consideration of the de 
velopement of the radical, that (3 is of this form 



m. . m 

If we substitute this value in the differential equation in J3 9 the compari 
son of like powers of m will give 

A l (i-n-2s+2).(i-n-2 S + 1) s _ 

2 s (2 i 2 s + 1) 

whence we derive, by successively putting s = 1, s = 2, &c. the values of 
A (l) , A (2) , and consequently, 







/ 
( 



__ 

n-n--n--n--n--n- 

2.4.6(2i l)(2i 3)(2i 5) 

A is a function of m independent of m ; but m and m entering alike into 
the preceding radical, they ought to enter similarly into the expression of 
13 ; we have therefore 



7 being a coefficient independent of m and m ; therefore 

. 1 m 



Thus we see that /3 is split into three factors, the first independent of 
m and m ; the second a function of m alone ; and the third a like function 
of m. We have only now to determine 7. 

For that purpose, we shall observe, that if i n be even ; we have, 
supposing m = 0, and m = 0, 

A _ y.U.2....i n} 2 _ 
= [2. 4. . . . (i n). (2 i 1). (2 i 3). . . . (i + n+ I)} 2 

7. U- 3. 5....(i n1). 1.8.5.... (i + nl)j 
U.3. 5.... (2 i !)} 



240 A COMMENTARY ON [SECT. XII. & XIII. 

If i n is odd, we shall have, in retaining only the first power of m, 
and m , 

_ y.m.m {1. 2.... (i n)}* _ 



_ 
" [2. 4 ---- (i n 1) (2i 1) (2i 3).. 

_ y.m. m H. 3. 5.... (i n). 1.3. 5.... (i + n)} 2 

U- 3. 5 ____ (2 i 1)1 2 
The preceding radical becomes, neglecting the squares of m, m , 

{r*-2 R r cos.( w -*/)+ R 2 }~* + R r. m nr {r*-2r R cos. (- ) + R 2 }~ ; . (f ) 
If we substitute for cos. (a ar ), its value in imaginary exponentials, 
and if we call c the number whose hyperbolic logarithm is unity, the part 
independent of m m , becomes 

{r R.c(- -OV 31 " 1 }"*. [r B.c^(*- )v^=I]-^. 

The coefficient of 

Ri c n( w -OV-l + c _n( w - w )V-l R 1 

TTTT- - 2 - or of rT+ri cos - n ( w w ) 
in the developement of this function is 

2. 1. 3. 5 ____ (i + n 1). 1. 3. 5 ____ (i n 1) 

2. 4. 6 ---- (i + n) 2. 4. 6 ____ (i n) 

This is the value of /3 when i n is even. Comparing it with that 
which in the same case we have already found, we shall have 

/I. 3. 5. ...(2i l)x* i(i l)....(i n+ 1) 
" \ 1.2.3 ---- i / ^(i+l)(i+2) ---- (i + n) 
When n = 0, we must take only half this coefficient, and then we 

have 

_ /I. 3. 5 ____ 2i K 2 
7 := \ 1. 2. 3 ____ i / 

R 1 

In like manner, the coefficient of - , . , m . m cos. n (* & } in the 

r + i 

function (f) is 

2. 1. 3. 5 ____ (i + n) . 1. 3. 5 ---- (i n) 



2. 4. 6. (i + n 1) . 2. 4. 6 (i n 1) 

this is the coefficient of m m in the value of /3, when we neglect the 
squares of m, m , and when i n is odd. Comparing this with the va 
lue already found, we shall have 

> /I- 3. 5 (2i IK* i(i l)....(i n + 1. 

V V 1.2.3 i ) (i+1) (i + 2) (i + n) 

an expression which is the same as in the case of i n being even. 
If n = 0, we also have 

/I. 3. 5.... (2 i- IV 
7 \ 1.2.3 i ) 



BOOK I.] NEWTON S PRINCIPIA. 241 

560. From what precedes, we may obtain the general form of functions 
Y w of m, V 1 m 2 . sin. -a, and V 1 m 2 . cos. r, which satisfy the 
equation of partial differences 




= - - . 

\ dm / 1 m 2 

Designating by /3, the coefficient of sin. n &, or of cos. n ^, in th 
function Y (1) , we shall have 



- - r - -T. . 

dm 1 m 2 

8 is equal to (I m 2 ) & multiplied by a rational and entire function of m, 
and in this case, by the preceding No., we have 



A (n) being an arbitrary constant ; thus the part of Y (i > depending on the 
angle n , is 



+ B (n cos. n *?} ; 

A (n) and B (n) being two arbitraries. If we make successively in this func 
tion, n = 0, n = 1, 11 = 2 . . . n = i ; the sum of all the functions which 
thence result, will be the general expression of Y (l) , and this expression 
will contain 2 i + 1 arbitraries B c >, A >, B <, A , B < 2 >, &c. 

Let us now consider a rational and entire function S of the order s, 
of the three rectangular coordinates x, y, z. If we represent by R the 
distance of the point determined by these coordinates from their origin ; 
by 6 the angle formed by R and the axis of x ; and by -a the angle which 
the plane of x, y forms with the plane passing through R and the axis of 
x ; we shall have 



x = Rm;y = R. VI m z . cos. ; z = R V 1 m 2 . sin. . 

Substituting these values in S, and developing this function into sines 
and cosines of the angle -a and its multiples, if S is the most general func 
tion of the order s, then sin. n w, and cos. n *r, will be multiplied by func 
tions of the form 

n 

(1 _ m s ) MA .m s - n + B.m 8 -"- 1 -f C.m s ~ n - 2 + &c.}; 

thus the part of S, depending on the angle n , will contain 2 (s n-f-1) 

indeterminate constants. The part of S depending on the angle ^ and its 

multiples will contain therefore s (s + 1) indeterminates; the part inde- 

VOL. II. Q, 



242 A COMMENTARY ON [SECT. XII. & XIII. 

pendent of will contain s + 1, and S will therefore contain (s + 1) * 
indeterminate constants. 

The function Y (0 > + Y (1 ) + &c. Y (s > contains in like manner (s + 1) * 
indeterminate constants, since the function Y (i) contains 2 i + 1 ; we may 
therefore put S into a function of this form, and this may be effected as 
follows : 

From what precedes we shall learn the most general expression of Y (s) , 
we shall take it from S and determine the arbitraries of Y (s) so that the 



powers and products of m and V 1 m 2 of the order s shall disappear 
from .the difference S Y (s) ; this difference will thus become a function 
of the order s 1 which we shall denote by S . We shall take the most 
general expression of Y (s - 1} ; we shall subtract it from S , and determine 
the arbitraries of Y^"" 15 so that the powers and products of m and 
V 1 m 2 of the order s 1 may disappear from the difference 
S Y (s-1) . Thus proceeding we shall determine the functions Y (s) , 
Y< s - 1 ), Y (s - 2) , &c. of which the sum is S, 

561. Resume now, the equation of No. 559, 

U =f s . R i + 2 d R . d m . d */. Q . 

Suppose R a function of m , -a and of a parameter a, constant for all 
shells of the same density, and variable from one shell to another. The 
difference d R being taken on the supposition that m , */ are constant we 
shall have 



therefore 

a d m d " Q " 



Let R i + 3 be developed into a series of the form 

Z W + Z W + Z + &c, 

Z (i) being whatever i may be, a rational and entire function of m , 
^/ 1 _ nv 7 ^. s i n . w } and VI m 2 . cos. & , which satisfies the equation 
of partial differences 




The difference of Z (i) taken relatively to a, satisfies also this equation, 
and consequently it is of the same form ; by the general theorem of 556, 
we ought therefore only to consider the term Z (i > in the developement of 
R i + 3 , and then we have 



BOOK I.] k NEWTON S PRINCIPIA. 243 

When the spheroid is homogeneous and differing but little from a 
sphere, we may suppose g = 1, and R = a ( 1 + a y ) ; then we have, by 
integrating relatively to a 

U co = L /Z . d m . d */. Q . 
1 + 

Moreover, if we suppose y developed into a series of the form 

Y + Y /(1 > + Y + &c., 

Y (i) satisfying the same equation of partial difference as Z (i) ; we shall have, 
neglecting quantities of the order a 2 , Z w: = (i + 3). a. a i + 3 Y /(i) ; we 
shall therefore have 

U = a . a 4 + 3 ./ Y . d m . d V. Q . 

If we denote by Y (i) what Y (i) becomes when we change m and -a into 
m and -a ; we shall have by No. 554, 



U - --L Y- 
2 i + 1 

we therefore have this remarkable result, 

4 v Y (i) 



(1) 



This equation subsisting whatever may be Y (i) we may conclude ge 
nerally that the double integration of the function f Z (i) d m . d . Q W 
taken from m = 1 to m = 1, and from */ = to * = 2 T, only 

4 v Z (i) 
transforms Z (i) into =-. =- ; Z (i) being what Z (i) becomes when we 

change m and * into m and -a ; we therefore have 

4 * , /d Z (i \ . 

= - d a; 



and the triple integration upon which U (i) depends, reduces to one in 
tegration only taken relatively to a, from a = to its value at the surface 
of the spheroid. 

The equation (1) presents a very simply method of integrating the func 
tion f Y (i) . Z (i) . d m . d 9, from m = 1 to m = 1, and from -a 
to = 2 it. In fact, the part of Y depending on the angle n v, is by 
what precedes, of the form X {A^ sin. n + B (n) cos. n *}, X being 
equal to 

(1 __ m -*)f - {mi-n-i 1 - 11 ^ 1 .^- 1 ) . m^ ^ + &c. }; 

we shall have therefore 

Y (i) = X {A (n) sin. n *> + B (n ) cos. n *S\ ; 

X 7 being what X becomes when m is changed into m . The part of Q (l) 
depending on the angle n *, is by the preceding No., y X X cos. n (*r ), 



244 A COMMENTARY ON [SECT. XII. & XIII. 

or 7 X . X.cos. n a. cos. n / + sin. n . sin. n & } ; thus that part of the 
integral/ Y . d m . d . Q 0) which depends on the angle n , will be 
7 A. sin. n ^./X 2 . d m . d */. sin. n */ {A < n > sin. n + B W cos. n </} 
7 X. cos. n w/x 2 . d m . d -at. cos. n JA (n ^ sin. n ^ + B (n > cos. n w }. 
Executing the integrations relative to = , that part becomes 

7 X -a [A (n) sin. n * + B (n) cos. n wj./x 2 . d m ; 
but in virtue of equation (1), the same part is equal to 

4<r 
X-T ; r . X. f A (n > sin. n w + B (n > cos. n r? 



Now represent by X A /(n > sin. n + B /(n > cos. n sr] that part of Z (i) 
which depends on the angle n *. This part ought to be combined with 
the corresponding part of Y ^ ; because the terms depending on the sines 
and cosines of the angle and its multiples, disappear by integration, in 
the function/ Y (i) Z (i) d m . d , integrated from * to -a = 2 *; we 
shall thus obtain, in regarding only that part of Y (i > which depends on 
the angle n w, 

/Y W. Z W d m d * = 
/X 2 . d m . d *{ A W sin. n + B (n ) cos. n *,} {A. < n > sin. n 9 + B (n ) cos. n *} 

B 7 ^)}. Adm=. . 4<r 



Supposing therefore successively in the last member n = 0, n = 1, 
n = 2 . . . n = i; the sum of all the terms, will be the value of the in 
tegral/ Y ZWdm.dw. 

If the spheroid is one of revolution, so that the axis with which the ra 
dius R forms the angle w, may be the axis of revolution ; the angle -a will 
disappear from the expression of Z (i) , which then takes the following 
form: 

1.3.5. ..2i 1 n f n i. (i-1) , i.(i-l) (i-2) (i-3> . 

W < m 03 > _ L. m ~ 2 -I- - - - - -m 1 "* ??c 
( 2.2i-l + 2.4.2 C 



__ - _ 
1.2.3. ..i 

A (i ) being a function of a. Call X W the coefficient of A (i) , in this func 
tion : the product 

/1.3.5...(2i IK 2 ( . i. (i 1) 1 s 

( 1.2.3.. .i )-i 1 -a.(8i-i)-+* c } 

R 

is by the preceding No., the coefficient of j in the developement of 

the radical 



2 2 R r {m m + V 1 m 2 . V 



cos. 



BOOK I.] NEWTON S PIUNCIPIA. 245 

when we therein suppose m and m equal to unity. This coefficient is 
then equal to 1 ,- we have therefore 

I.3.5...(2i-l)( i (i - 1) 1 _ 

1. 2. 3...i V 2 (2 i lp J" 
that is to say, X W reduces to unity, when m = 1. We have then 



_ _ 

" (i + 3). (2i + Iy da 

Relatively to the axis of revolution, m = 1, and consequently, 

4 it /d A (i) 



therefore if we suppose that relatively to a point placed upon this axis 
produced, we have 



we shall have the value of V relative to another point placed at the mean 
distance from the origin of coordinates, but upon a radius which makes 
with the axis of revolution, an angle whose cosine is m ; by multiplying 
the terms of this value respectively by X c % X W, X ( % &c. 

In the case when the spheroid is not of revolution, this method will 
give the part of V independent of the angle -a : we shall determine the 
other part in this manner. Suppose for the sake of simplicity, the sphe 
roid such that it is divided into two equal and similar parts by the equa 
tor, whether by the meridian where we fix the origin of the angle &, or 
by the meridian which is perpendicular to the former. Then V will be 
a function of m 2 , sin. 2 w, and cos. 2 , or which comes to the same, it will 
be a function of m 2 , and of the cosine of the angle 2 -a and its multiples ; 
U (l) will therefore be nothing, when i is odd, and in the case when it is 
even, the term which depends on the angle 2 n v, will be of the form 

C . (l-mWm - "*"|<?- 2n 1 - 1) m -- +&c.}cos. 2 n ,. 

^- * (/& 1 Jij J 

Relatively to an attracted point placed in the plane of the equator, 
where m = 0, that part of V which depends on this term becomes 
+ C) _ 1. 3. 5...(i 2n 1) __ 
r r i + 1 2 (i + 2n + l)(i + 2n + 2)...(2i 1) * 
whence it follows that having developed V into a series ordered according 
to the cosines of the angle. 2 & and its multiples, when the point attracted 
is situated in the plane of the equator ; to extend this value to any attract 
ed point whatever, it will be sufficient to multiply the terms which depend 

COS. 2 n nr 

on -r - by the junction 

Q3 



216 A COMMENTARY ON [SECT. XII. & XIII. 



_ 
-1.3.5. ..(i 2n I) M ~2~(2 i 1) 

m i_2n- 2+ 

we shall hence obtain, therefore, the entire value of V, when this value 
shall be determined in a series, for the two cases where the part attracted 
is situated upon the polar axis produced, and where it is situated in the 
plane of the equator; this greatly simplifies the research of this value. 

The spheroid which we are considering comprehends the ellipsoid. 
Relatively to an attracted point situated upon the polar axis, which we 
shall suppose to be the axis of x, by 546, we have b = 0, c 0, and 
then the expression of V of No. 549, is integrable relatively to p. Rela 
tively to a point situated in the plane of the equator, we have a = 0, and 
the same expression of V still becomes, by known methods, integrable re 
latively to q, by making tan. q = t. In the two cases, the integral being 
taken relatively to one of these variables in its limits, it then becomes 
possible relatively to the other, and we find that M being the mass of 

V 

the spheroid, the value of ^ is independent of the semi-axis k of the 

spheroid perpendicular to the equator, and depends only on the ex- 
centricities of the ellipsoid. Multiplying therefore the different terms 

V 

of the values of ^ relative to these two cases, and reduced into se 

ries proceeding according to the powers of - , by the factors above men- 

y 

tioned, to get the value of -r-=. relative to any attracted point whatever; the 

function which thence results will be independent of k, and only depend 
on the excentricities ; this furnishes a new demonstration of the theorem 
already proved in 550. 

If the point attracted is placed in the interior of the spheroid, the at 
traction which it undergoes, depends, as we have seen in No. 553, on the 
function v (i; , and by the No. cited, we have 
r ?A Rdm dV. Q 

TT \l) / *L -- 

-J R 1 - 1 

an equation which we can put under this form 

d a - d m/ - d "- Q(i) - 



Suppose R 2 - 1 developed into a series of the form 

z (0) + Z U) + Z (2) + C . 



BOOK I.] NEWTON S PR1NCIPIA. -. 247 

z (i > satisfying the equation of partial differences, 

j / / - 
dl(l 

( 




if moreover we call z w what z (i) becomes when we change m into m, and 
v into -0) we shall have by what precedes, 
4r . >dz (i 



thus therefore we shall get the expression of V relative to all the shells of 
the spheroid which envelope the point attracted. The value of V relative 
to shells to which it is interior, we have already shown how to deter 
mine. 



ON THE FIGURE OF A FLUID HOMOGENEOUS MASS IN EQUILIBRIUM, 
AND ENDOWED WITH A ROTATORY MOTION. 

562. Having exposed in the preceding Nos. the theory of the attrac 
tions of spheroids, we now proceed to consider the figure which they 
must assume in virtue of the mutual action of their parts, and the other 
forces which act upon them. We shall first seek the figure which satis 
fies the equilibrium of a fluid homogeneous mass endowed with a rotatory 
motion, and of that problem we shall give a rigorous solution. 

Let a, b, c be the rectangular coordinates of any point of the surface of 
the mass, and P, Q, R the forces which solicit it parallel to the coordi 
nates, the forces being supposed as tending to diminish them. We know 
that when the mass is in equilibrium, we have 

= P. da + Q. db + R. d c; 

provided that in estimating the forces P, Q, R, we reckon the centrifugal 
force due to the motion of rotation. 

To estimate these forces, we shall suppose that the figure of the fluid 
mass, is that of the ellipsoid of revolution, whose axis of rotation, is the axis 
itself of revolution. If the forces P, Q, R which result from this hypothe 
sis, substituted in the preceding equation of equilibrium give the differen 
tial equation of the surface of the ellipsoid ; the preceding hypothesis is 
legitimate, and the elliptic figure satisfies the equilibrium of the fluid 
mass. 

Suppose that the axis of a is that also of revolution ; the equation of 
the surface of the eUipsoid will be of this form 
a 2 + m (b 2 + c 2 ) = k s ; 
Ql 



248 A COMMENTARY ON [SECT. XII. & XIII. 

the origin of the coordinates a, b, c being at the center of the ellipsoid, 
k will be the semi-axis of revolution, and if we call M the mass of the el 
lipsoid, by 546, we shall have 



3 m 

g being the density of the fluid. If we make as in 547, m = X 2 , we 

shall have m = , and consequently 

* "* 



. 

an equation which will give the semi-axis k, when X is known. 
Let 



B/ = ^* (1 +* 2 )tan.- X X)}; 
we shall have by 547, regarding only the attraction of the fluid mass 

P = A a; Q = B b; R = B c. 

If we call g, the centrifugal force at the distance 1, from the axis of 
rotation ; this force at the distance V b " + c 2 from the same axis, will 
be g V b 2 -f- c s : resolving this parallel to the coordinates b, c there will 
result in Q the term g b, and in R the term g c; thus we shall have, 
reckoning all the forces which animate the molecules of the surface, 

P = A a; Q = (B -g)b; R = (B g). c; 
the preceding equation of equilibrium, will therefore become 

= a d a + B ~ g (b d b + c d c). 
The differential equation of the surface of the ellipsoid is by substitut 



in for m its value 



, 
X 

b d b -f- c d c 

- 



, 

= ad + x 

by comparing this with the preceding one, we shall have 

(1 + >- 2 )(B -g) = A ; ........ (1) 

if we substitute for A , B their values, and if we make r^ = q ; we shall 

7 V 
have 



BOOK I.J NEWTON S PRINCIPIA. 249 

determining therefore X by this equation which is independent of the co 
ordinates a, b, c, the equation of equilibrium will coincide with the equa 
tion of the surface of the ellipsoid ; whence it follows, that the elliptic fi 
gure satisfies the equilibrium, at least, when the motion of rotation is such 
that the value of X 2 is not imaginary, or when being negative, it is neither 
equal to nor greater than unity. The case where X 2 is imaginary would 
give an imaginary solid; that where X 2 = 1, would give a paraboloid, 
and that where X 2 is negative and greater than unity, would give a hy- 
perboloid. 

563. If we call p the gravity at the surface of the ellipsoid, we shall 
have 

p = V P 2 + Q 2 + R 2 . 

In the interior of the ellipsoid, the forces P, Q, R, are proportional to 
the coordinates a, b, c ; for we have seen in No. 547, that the attractions 
of the ellipsoid, parallel to these coordinates, are respectively proportional 
to them, which equally takes place for the centrifugal force resolved pa 
rallel to the same coordinates. Hence it follows, that the gravities at dif 
ferent points of a radius drawn from the center of the ellipsoid to its sur 
face, have parallel directions, and are proportional to the distances from 
the center ; so that if we know the gravity at its surface, we shall have 
the gravity in the interior of the spheroid. 

If in the expression of p, we substitute for P, Q, R, their values given 
in the preceding No., we shall have 

p = V A 2 a 2 + (B -g) 2 . (b 8 + c 2 ); 
whence we derive, in virtue of equation (1) of the preceding No. 



p = 

-^ ^x -r /v ; 

b 2 + c 2 
but the equation of the surface of the ellipsoid gives -, $ = k 2 a 2 ; 

1 -j~ A 

we shall therefore have 

/ k * + X 2 a~* 
A V 1 + x 2 

a is equal to k at the pole, and it is nothing at the equator ; whence it fol 
lows, that the gravity at the pole is to the gravity at the equator, as 
V 1 + X 2 is to unity, and consequently, as the diameter of the equatoi 
is to the polar axis. 

Call t the perpendicular at the surface of the ellipsoid, produced to 
meet the axis of revolution, we shall have 

t = V (1 + X 2 ) (k 2 + X a 2 ) ; 



250 A COMMENTARY ON [SECT. XII. & XIII. 

wherefore 

A t 
= 



1 + A 2 
thus gravity is proportional to t. 

Let 4- be the complement of the angle which t makes with the axis of 
revolution ; 4 will be the latitude of the point of the surface, which we 
are considering, and by the nature of the ellipse, we shall have 






V 1 + A 2 cos. 2 <4/ 
we therefore shall have 

_ A k 

V I + X 2 . cos. 2 4 
and substituting for A its value, we shall get 

4rg.k.(l + X 2 ). (X tan.- X) . . 

X 3 V 1 + X 2 . cos. 2 4 

this equation gives the relation between gravity and the latitude ; but we 
must determine the constants which it contains. 

Let T be the number of seconds in which the fluid mass will effect a 
revolution ; the centrifugal force at the distance 1 from the axis of revo- 

4 7T 2 

lution, will be equal to -y ; we therefore have 

g 12 g 2 

q== f*..e- 4* f T 25 
which gives 

12. cr 2 

4 K P 



q. 1 
The radius of curvature of the elliptic meridian is 

(l+A 2 )k , 

( 1 + X 2 cos. 2 4<) 2 
calling therefore c the length of a degree at the latitude -4y\v e shall have 

i + X * = = 200 c. 

/ 1 i > 2 2 J/\ f 

This equation combined with the preceding one, gives 

4?rg (1 + Xj^_ _ 20() c ^] + X 2 cos .s ^ l 2 ^. 

V 1 + A 2 cos. 2 4/ q J- 

thus we shall have 

A tan. - A 12 v 

q" 

Let 1 be the length of the simple pendulum which oscillates seconds ; 



p = 200 C(l + X 2 COS. 2 4) - 3 .-7TT5. 

A Cl JL 



BOOK!.] NEWTON S PRINCIPIA. 231 

from dynamics it results that p = -r 2 1 (seeX.) ; comparing these two 
expressions of p, we get 

- 2400 c (A tan.- 1 *.) (1 + X 2 cos. 2 ^) . 

*1T 2 A 3 

this equation and equation (2) of the preceding No. will give the values 
of q and X by means of the length 1 of the seconds pendulum, and the 
length c of the degree of the meridian, both being observed at the lati 
tude -^. 

Suppose 4> = 50, these equations will give 



. 800c .i 

*1T 2 4 WlT 



observations give, as we shall see hereafter, 
c = 100000;!= 0.741608; 

moreover we have T = 99727 ; we shall thus obtain 
q = 0.00344957 ; X 2 = 0.00868767. 

The ratio of the axis of the equator to the polar axis, being V 1 + X 2 , 
it becomes in this case 1.00433441 ; these two axes are very nearly in 
the ratio of 231.7 to 230.7, and by what precedes, the gravities at the 
pole and at the equator are in the same ratio. 

We shall have the semi polar axis k, by means of the equation 



200 c (1 + ^ 2 )* i 

*(1 + X 2 ) ~^~ {i 

which gives 

k = 6352534. 

To get the attraction of a sphere, whose radius is k. and density any 
whatever ; we shall observe that a sphere, having the radius k and density 
f, acts upon a point placed at its surface, with a force equal to f is g . k, 

and consequently, in virtue of equation (3) equal to rTT> 

o ( 1 -j- A j (X tan. X) 

/ o 

or to p (l -_ x 2 + &c.V or finally to 0.998697. p, p being the gravi 

ty upon the parallel of 50. Hence it is easy to obtain the attractive force 
of a sphere of any radius and density whatever, upon a point placed with 
in or without it. 

564. If the equation (2) of No. 562, were susceptible of many real 
roots, many figures of equilibrium might result from the same motion of 
rotation ; let us examine therefore whether this equation has several real 



252 A COMMENTARY ON [SECT. XII. & XIII. 

0X1 2 Q X 3 

roots. For that purpose, call <p the function - - 5^ -- tan.- l X, 

y "I* o A 

which being equated to zero, produces the equation (2). It is easy to see, 
that by making X increase from zero to infinity, the expression of <p begins 
and ends by being positive ; thus, by imagining a curve whose abscissa is 
X and ordinate p, this curve will cut its axis when X = ; the ordinates 
will afterwards be positive and increasing ; when arrived at their maxi 
mum, they will decrease; the curve will cut the axis a second time at a 
point which will determine the value of X corresponding to the state of 
equilibrium of the fluid mass; the ordinates will then be negative, and 
since they are positive when X = oo ; the curve necessarily cuts the axis 
a third time, which determines a second value of X which satisfies the 
equilibrium. Thus we see, that for one and the same value of q, or for 
one given motion of rotation, there are several figures for which the 
equilibrium may subsist. 

To determine the number of these figures, we shall observe, that we 
have 

_ 6 X g dXJq X 4 + (10 q 6) X*+ 9 q} 
(3 X 2 +9) 2 . (1 + X 2 ) 

The supposition of d <p = 0, gives 

= qX* + (10 q 6) X 2 + 9 q; 
whence we derive, considering only the positive values of X 



These values of X determine the maxima and minima of the ordinate <p ; 
there being only two similar ordinates on the side of positive abscissas, on 
that side the curve cuts its axis in three points, one of them being the 
origin ; thus, the number of figures which satisfy the equilibrium is reduc 
ed to two. 

The curve on the side of negative abscissas being exactly the same as 
on the side of positive abscissas ; it cuts its axis on each side in corre 
sponding points equidistant from the origin of coordinates ; the negative 
values of X which satisfy the equilibrium, are therefore, as to the sign 
taken, the same as the positive values ; which gives the same elliptic fi 
gures, since the square of X only enters the determination of these figures ; 
it is useless therefore to consider the curve on the side of negative ab 
scissas. 

If we suppose q very small, which takes place for the earth, we may 
satisfy equation (2) of 562, in the two hypotheses of X 2 being very small, 



BOOK L] NEWTON S PRINCIPIA. 253 

and of X 2 being very great. In the first, by the preceding No., we 
have 



To get the value of X 2 in the second hypothesis, we shall observe that 

CT 

then tan." 1 X differs very little from *, so that if we suppose X = - , 

a will be a very small angle of which the tangent is - ; we shall there 
fore have, p. 27. Vol. I. 



1 

a= x 

and consequently 



equation (2) of No. 562, will thus become^ 

9X+2q.X*__ ff 1 . j_ 
9 + 3X 2 "2 X^3X 3 
whence by the reversion of series we get 

3cr 8 4 q /. 64 N 
X = - ---- -- M 1 - 2} + &c. 

4 T T \ 3 cr v 



- 

4 q T T \ 3 cr 

= 2.356195. -L 2.546479 1.478885 q + &c. 

We have seen in the preceding No., that relatively to the earth, 
q = 0.00344957 ; this value of q substituted in the preceding expression, 
gives X = 680.49. Thus the ratio of the two axes equatorial and polar, 
a ratio which is equal to V 1 + X 2 , is in the case of a very thin spheroid, 
equal to 680.49. 

The value of q has a limit beyond which the equilibrium is impossible, 
the figure being elliptic. Suppose, in fact, that the curve cuts its axis 
only at its origin, and that in the other points it only touches; at the 
point of contact we shall have <p = 0, and d <p = ; the value of p will 
never therefore be negative on the side of positive abscissas, which are 
the only ones we shall here consider. The value of q determined by the 
two equations p = 0, d p = 0, will therefore be the limit of those with which 
the equilibrium can take place, so that a greater value will render the 
equilibrium impossible ; for q being supposed to increase by f, the func- 

2 f X 3 
tion <p increases by the term jr ^ ; thus, the value of <p correspond- 

*s "^ o A 

ing to q, being never negative, whatever X may be, the same function cor 
responding to q + fj is constantly positive, and can never become no- 



254 A COMMENTARY ON [SECT. XII. & XIII. 

thing ; the equilibrium is then therefore impossible. It results also from 
this analysis, that there is only one real and positive value of q, which 
would satisfy the two equations <p = 0, and d <p 0. These equations 
give 



7 X 5 + 30 X 3 + 27 X 
= - 



_ 
= (1 + X 2 ) (9 + X 2 ) 

7 X 5 + 30 X 3 + 
(1 + x 2 )(3 + X*) 

The value of X which satisfies this last equation is X = 2.5292 ; whence 
we get q = 0.337007 ; the quantity V 1 -J- X 2 , which expresses the ra 
tio of the equatorial axis to the polar axis, is in this case equal to 2.7197. 

The value of q relatively to the earth is equal to 0.00344957. This 
value corresponds to a time of rotation of 0.99727 days ; but we have 

generally q = r** so that relatively to masses of the same density, q is 

o 5 

proportional to the centrifugal force g of the rotatory motion, and conse 
quently in the inverse ratio of the square of the time of rotation ; whence 
it follows, that relatively to a mass of the same density as the earth, the 
time of rotation which answers to q = 0.337007, is 0.10090 days. Whence 
result these two theorems. 

" Every homogeneous fluid mass of a density equal to the mean density 
of the earth, cannot be in equilibrium having an elliptic figure, if the time 
of its rotation is less than 0.10090 days. If this time be greater, there 
will be always two elliptic figures and no more which satisfy the equili 
brium." 

" If the density of the fluid mass is different from that of the earth ; we 
shall have the time of rotation in which the equilibrium ceases to be pos 
sible under an elliptic figure, by multiplying 0.10090 days by the square 
root of the ratio of the mean density of the earth to that of the fluid 
mass." 

This relatively to a fluid mass, whose density is only a fourth part of 
that of the earth, which nearly is the case with the sun, this time will be 
0.20184 days; and if the density of the earth supposed fluid and homo 
geneous were about 98 times less than its actual density, the figure which 
it ought to take to satisfy its actual motion of rotation, would be the limit 
of all the elliptic figures with which the equilibrium can subsist. The 
density of Jupiter being about five times less than that of the earth, and 
the time of its rotation being 0.41377 days; we see that this duration is 
in the limits of those of equilibrium. 



BOOK I.] NEWTON S PRINCIPIA. 

It may be thought that the limit of q, is that where the fluid would be 
gin to fly off by reason of a too rapid motion of rotation ; but it is easy to 
be convinced of the contrary, by observing that by 563, the gravity at the 
equator of the ellipsoid is to that at the pole in the ratio of the polar axis 
to that of the equator, a ratio which in this case, is that of 1 to 2.7197 ; 
the equilibrium ceases therefore to be possible, because with a motion of 
rotation more rapid, it is impossible to give to the fluid mass, an elliptic 
fio-ure such that the resultant of its attraction and of the centrifugal force, 
may be perpendicular to the surface. 

Hitherto we have supposed X 2 positive, which gives the spheroids flat 
tened towards the poles ; let us now examine whether the equilibrium can 
subsist with a figure lengthened towards the poles, or with a prolate sphe 
roid. Let X 2 = X /2 ; X 2 must be positive and less than unity, otherwise, 
the ellipsoid will be changed into a hyperboloid. The preceding value 
of d p gives 

x.X 2 dxqX 4 + (10 q 6) X g + 9 qj _ 
^ ~ J ~ (1 + X 2 ) (9 + 3 X 2 ) 2 

the integral being taken from X = 0. Substituting for X its value + X V - 1, 
we shall have 



- - - 

l J (l _x /2 ) (9 3 X 2 ) 

but it is evident that the elements of this last integral are all of the same 
sign from X /2 = 0, to X /2 = 1 ; the function p can therefore never be 
come nothing in this interval. Thus then the equilibrium cannot subsist 
in the case of the prolate spheroid. 

565. If the motion of rotation primitively impressed upon the fluid 
mass, is more rapid than that which belongs to the limit of q, we must 
not thence infer that it cannot be in equilibrium with an elliptic figure ; 
for we may conceive, that by flattening it more and more, it will take a 
rotatory motion less and less rapid ; supposing therefore that there exists, 
as in the case of all known fluids, a force of tenacity between its mole 
cules, this mass, after a great number of oscillations, may at length arrive 
at a rotatory motion, comprised within the limits of equilibrium, and may 
continue in that state. But this possibility it would also be interesting to 
verify ; and it would be equally interesting to know whether there would 
not be many possible states of equilibrium ; for what we have already de 
monstrated upon the possibility of two states of equilibrium, correspond 
ing to one motion of rotation, does not infer the possibility of two states 
of equilibrium corresponding to one primitive force ; because the two 



256 A COMMENTARY ON [SECT. XII. & XIII. 

states of equilibrium relative to one motion of rotation, require two pri 
mitive forces, either different in quantity or differently applied. 

Consider therefore a fluid mass agitated primitively by any forces what 
ever,- and then left to itself, and to the mutual attractions of all its parts. 
If through the center of gravity of this mass supposed immoveable, we 
conceive a plane relatively to which the sum of the areas described upon 
this plane, by each molecule, multiplied respectively by the correspond 
ing molecules, is a maximum at the origin of motion ; this plane will 
always have this property, whatever may be the manner in which the 
molecules act upon one another, whether by their tenacity, by their attrac 
tion, and their mutual collision, even in the very case where there is finite 
loss of motion in an instant of time ; thus, when after a great number of 
oscillations, the fluid mass shall take a uniform rotatory motion about a 
fixed axis, this axis shall be perpendicular to the plane above-mentioned, 
which will be that of the equator, and the motion of rotation will be such 
that the sum of the areas described during the instant d t, by the mole 
cules projected upon this plane, will be the same as at the origin of mo 
tion ; we shall denote by E d t this last sum. 

We shall here observe, that the axis in question, is that relatively to 
which the sum of the moments of the primitive forces of the system was a 
maximum. It retains this property during the motion of the system, and 
finally becomes the axis of rotation ; for what is above asserted as to the 
plane of the maximum of projected areas, equally applies to the axis of the 
greatest moment of forces ; since the elementary area described by the pro 
jection of the radius-vector of a body upon a plane, and multiplied by its 
mass, is evidently proportional to the moment of the finite force of this 
body relatively to the axis perpendicular to this plane. 

Let, as above, g be the centrifugal force due to the rotatory motion at 
the distance 1 from the axis; V g will be the angular velocity of rotation 
(p. 166. Vol. I.) ; then call k the semi-axis of rotation of the fluid mass, 
and k V 1 + A z the semi-axis of its equator. It is easy to show that 
the sum of the areas described during the instant d t, by all the molecules 
projected upon the plane of the equator and multiplied respectively by the 
corresponding molecules, is 

1(1 + A*) 2 .k 5 dt Vg 



we shall therefore have 



BOOK I.] NEWTON S PRINCIPIA. 257 

Then calling M, the fluid mass, we shall have 
$*k> ff (l + X 2 ) = M; 

the quantity r^ > which we have called q, in No. 562, thus becomes 

q (1 + X )~^ denoting by q the function 2 ^J_JLlf The equa 
tion of the same No. becomes 






9 + 3X 2 

This equation will determine X ; we shall then have k by means of the 
preceding expression of M. 
Call <p the function 

9 + 3X 2 tan ~~ l X> 

which, by the condition of equilibrium, ought to be equal to zero : this 
equation begins by being positive, when X is very small, and ends by being 
negative, when X is infinite ; there exists therefore between X = 0, and 
X = infinity, a value of X which renders this function nothing, and conse 
quently, there is always, whatever q may be, an elliptic figure, with which 
the fluid mass may be in equilibrium. 

The value of <p may be put under this integral form 

/X 4 dx{^-+ 18 q fq X 2 + 18(1 + X 2 ) 

<f> = 2 I L- 

(9 + 3 X 2 ) 2 (1 + X 2 )* 
When it becomes nothing the function 

? + 18q -fq X 2 + 18(1 + * 2 ) f ], 

has already passed through zero to become negative ; but from the in 
stant when this function begins to be negative, it continues to be so as X 

27 q 
increases, because the positive part f> + 18 q decreases whilst the ne- 

X 

gative part {q X 2 + 18 (1 + X 2 )S} increases; the function p cannot 
therefore twice become nothing ; whence it follows, that there is but one 
real and positive value of X which satisfies the equation of equilibrium, 
and consequently, the fluid can be in equilibrium with one elliptic figure 
only. 



Vor.. IT. R 



258 A COMMENTARY ON [SECT. XII. & XIII. 

ON THE FIGURE OF A SPHEROID DIFFERING VERY LITTLE FROM A SPHERE, 
AND COVERED WITH A SHELL OF FLUID IN EQUILIBRIUM. 

566. We have already discussed the equilibrium of a homogeneous 
fluid mass, and we have found that the elliptic figure satisfies this equili 
brium; but in order to get a complete solution of the problem, \ve must 
determine a priori all the figures of equilibrium, or we must be certified 
that the elliptic is the only figure which will fulfil these conditions; be 
sides, it is very probable that the celestial bodies have not homogeneous 
masses, and that they are denser towards the center than at the surface. 
In the research, therefore, of their figure, we must not rest satisfied with 
the case of homogeneity ; but then this research presents great difficul 
ties. Happily it is simplified by the consideration of the little difference 
which exists between the spherical figure and those of the planets and 
satellites; by which we are permitted to neglect the square of this differ 
ence, and of the quantities depending on it. Notwithstanding, the research 
of the figure of the planets is still very complex. To treat it with the 
greatest generality, we proceed to consider the equilibrium of a fluid mass 
which covers a body formed of shells of variable density, endowed with 
a rotatory motion, and sollicited by the attraction of other bodies. For 
that purpose, we proceed to recapitulate the laws of equilibrium of fluids, 
as laid down in works upon hydrostatics. 

If we name g the density of a fluid molecule, II the pressure it sustains, 
F, F , F", &c. the forces which act upon it, and d f, d f , d f " the ele 
ments of the directions of these forces; then the general equation of the 
equilibrium of the fluid mass will be 

- F d f + F d f + F" d f " + &c. 

S 
Suppose that the second member of this equation is an exact difference; 

designating by d p this difference, g will necessarily be a function of n and 
of <p : the integral of this equation will give <p in a function of n ; we may 
therefore reduce to a function of n only, from which we can obtain n in 
a function of p ; thus, relatively to shells of a given constant density, we 
shall have d n = 0, and consequently 

= F d f + F d f + F" d f" + &c. ; 

an equation which indicates the tangential force at the surface of those 
shells is nothing, and consequently, that the resultant of all the forces 
F, F , F", &c. is perpendicular to this surface ; so that the shells are 
spherical. 



BOOK I.] NEWTON S PUINCIPIA. 259 

The pressure n being nothing at the exterior surface, must there be 
constant, and the resultant of all the forces which animate each molecule 
of the surface is perpendicular to it. This resultant is wnat we call gravi 
ty. The conditions of equilibrium of a fluid mass, are therefore 1st, that 
the direction of gravity be perpendicular to each point of the exterior sur 
face : 2dly, that in the interior of the mass the directions of the gravity of 
each molecule be perpendicular to the surface of the shells of a constant 
density. Since we may take, in the interior of a homogeneous mass, such 
shells as we wish for shells of a constant density, the second of two pre 
ceding conditions of equilibrium, is always satisfied, and it is sufficient for 
the equilibrium that the first should be fulfilled ; that is to say, that the 
resultant of all the forces which animate each molecule of the exterior 
surface should be perpendicular to the surface. 

567. In the theory of the figure of the celestial bodies, the forces F, F , 
F", &c. are produced by the attraction of their molecules, by the centrifu 
gal force due to their motion of rotation, and by the attraction of distant 
bodies. It is easy to be certified that the difference F d f + F d f + &c. 
is there exact ; but we shall clearly perceive that, by the analysis which 
we are about to make of these different forces, in determining that part of 
the integral t /(F d f + F d f -f &c.) which is relative to each of them. 

If we call d M any molecule of the spheroid, and f its distance from the 

point attracted, its action upon this latter will be ^- . Multiplying this 

action by the element of its direction, which is d f, since it tends to 
diminish f, we shall have, relatively to the action of the molecule d M, 

/F d f = -p- ; whence it follows that that part of the integral /(F d f 

+ F d f + &c.), which depends on the attraction of the molecules of 
the spheroid, is equal to the sum of all these molecules divided by their 
respective distances from the molecule attracted. We shall represent this 
sum by V, as we have already done. 

We propose, in the theory of the figure of the planets, to determine 
the laws of the equilibrium of all their parts, about their common center of 
gravity; we must, therefore, transfer into a contrary direction to the mole 
cule attracted, all the forces by which this center is animated in virtue of 
the reciprocal action of all the parts of the spheroid; but we know 
that, by the property of this center, the resultant of all the actions upon 
tliis point is nothing. To get, therefore, the total effect of the attraction 

R 2 



260 A COMMENTARY ON [SECT. XII. & XIII. 

of the spheroid upon the molecules attracted, \ve have nothing to add 
to V. 

To determine the effect of the centrifugal force, we shall suppose the 
position of the molecule determined by the three rectangular coordinates 
x , y , z , whose origin we fix at the center of gravity of the spheroid. 
We shall then suppose that the axis of x 7 is the axis of rotation, and that 
g expresses the centrifugal force due to the velocity of rotation at the dis 
tance I from the axis. This force will be nothing in the direction of x 
and equal to g y and g z in the direction of y and of z ; multiplying, 
therefore, these two last forces respectively by the elements d y 7 , d z of 
their directions, we shall have ^ g (y 8 + z 2 ) for that part of the integral 
f (F d f + F d f + &c.), which is due to the centrifugal force of the 
rotatory motion. 

If we call, as above, r the distance of the molecule attracted from the 
center of gravity of the spheroid, 6 the angle which the radius r forms with 
the axis of x , and * the angle the plane which passes through the axis 
of x , and through the molecule, forms with the plane of x , y ; finally, if 
we make cos. 6 = m, we shall have 

x = r m ; y = r V 1 m 2 . cos. -a ; z = r V 1 m z . sin. -a ; 
whence we get 

ig(y /2 + z 2 ) = *g* (l m )- 
We shall put this last quantity under the following form : 

4gr igr(m i) 

to assimilate its terms to those of the expression V which are given in No. 
559; that is to say, to give them the property of satisfying the equation of 
partial differences 




in which Y (i) is a rational and entire function of m, V 1 m * . cos. * 

and VI m 2 sin. of the degree i ; for it is clear that each of the two 

terms g r * and \ g r 2 (m 2 ) satisfies for Y , the preceding 
equation. 

It remains now for us to determine that part of the integral 
/"(F d f + F 7 d f + &c.) which results from the action of distant bodies. 
Let S be the mass of one of these bodies, f its distance from the molecule 
attracted, and s its distance from the center of gravity of the spheroid. 
Multiplying its action by the element d f of its direction, and then inte- 



BOOK I.] NEWTON S PRINCIPIA. 261 

c 
grating we shall have -TT-. This is not the entire part of the integral 

/(F d f + F d f + &c.) which is due to the action of S; we have still 
to transfer, in a contrary direction to the molecule, the action of this body 
upon the center of gravity of the spheroid. For that purpose, call v the 
angle which s forms with the axis of x , and 4< the angle which the plane 
passing through this star and through the body S, makes with the plane of 

S 
x , y . The action of ^ of this body upon the center of gravity of the 

spheroid, resolved parallel to the axes of x , y , z , will produce the three 
following forms : 

S S . S . 

g cos. v; - sin. v cos. 4; sin. v sin. 4-. 

s^ s z s 2 

Transferring them in a contrary direction to the molecule attracted, 
which amounts toprefixing to them the sign , then multiplying them by 
the elements d x , d y , d z , of their directions, and integrating them, the 
sum of the integrals will be 

g 
-- j- .x cos. v + y sin. v. cos. 4 1 + z sin. v sin. -^\ + const. ; 

the entire part of the integral /(F d f + F d f + &c.), due to the ac 
tion of the body S, will therefore be 

S S 
-f -- -^i* c s. v + y sin. v cos. 4/ + z sin. v sin. -^} + const. ; 

and since this quantity ought to be nothing relatively to the center of gra 
vity of the spheroid, which we suppose immoveable, and that relatively to 
this point, f becomes s, and x , y , z , are nothing, we shall have 

const. = . 
s 

However, f is equal to 

J(s cos. v x ) 2 + (s. sin. v cos. ^ y ) 2 + (s sin. v sin. -4, z ) }; 
which gives, by substituting for x , y , z , their preceding values 
S_ = S _ 

^s * 2s rcos. v cos. 6 + sin. v sin. 6 cos. (<*~^~-^)~+~x*} 
If we reduce this function into a series descending relatively to powers 
of s, and if we thus represent the series, 



we shall have generally by 56 1 and 562, 

L3.5..(2i-l) f i(i-l) , i(i-l)(i-2)(l-8) ) ; 

1.2.3 ..... i I 2(2 i If h 2.4(2i ])(2l=3) a I ; 



262 A COMMENTARY ON [SECT. XII. .& XIII. 

3 being equal to cos. v cos. d + sin. v sin. 6 . cos. (^ 4) > it is evident 
that by 553, we have 

= 



so that the terms of the preceding have this property, common with those 
of V. This being shown, we have 

s s s 

-TT ---- r(x cos. v + y sin. v cos. 4> + z SU1 - v sin> "^) 

I S S i 

p (2) +7 P (3) +7^ p w + &c 

If there were other bodies S , S", &c. ; denoting by s , v , 4 / P (i) ; s", 
v", 4/ , P" W, &c. what we have called s, v, 4/, P (i) , relatively to the body 
S, we shall have the parts of the integral /(F d f + F d P + &c.) due 
to their action, by marking with one, two, &c. dashes, the letters s, v, 4^ 
and P in the preceding expression of that part of this integral, which is 
due to the action of S. 

If we collect all the parts of this integral, and make 

J-=aZ>; 



&C. 

a being a very small coefficient, because the condition that the spheroid is 
very little different from a sphere, requires that the forces which produce 
this difference should themselves be very small ; we shall have 
/(Fdf + Fdf + &c.) = V + ar {Z>+ Z+ rZ+ r Z^ + &c.{ 
Z w satisfying, whatever i maybe, in the equation of partial differences 

d 8 Z 




m -, 

dm 



- JJ dm J .A q ~ x + i (i + J) Z<. 

\~ ~~d~5T~ J[. 1 m 2 

The general equation of equilibrium will therefore be 

f^JL = V + a r 2 {Z (0 > + Z< 2 > -f- r Z r 2 Z^ + &c.} . (1) 

o 

If the extraneous bodies are very distant from the spheroid, we may ne 
glect the quantities r * Z (3) , r 4 Z (4 >, &c., because the different terms of these 
quantities being divided respectively by s 4 , s 3 , &c. s /4 , s 3 , &c. these terms 
become very small when s, s , &c. are very great compared with r. This 



BOOK I.] NEWTON S PRINCIPIA. 263 

case subsists for the planets and satellites with the exception of Saturn, 
whose ring is too near his surface for us to neglect the preceding terms. 
In the theory of the figure of that planet, we must therefore prolong the 
second member of equation (1), which possesses the advantage of forming 
a series always convergent; and since then the number of corpuscles ex 
terior to the spheroid is infinite, the values of Z<>, Z, &c. are given in 
definite integrals, depending on the figure and interior constitution of the 
ring of Saturn. 

568. The spheroid may be entirely fluid ; it may be formed of a solid 
nucleus covered by a fluid. In both cases the equation (1) of the preced 
ing No. will determine the figure of the shells of the fluid part, by con 
sidering, that since n must be a function of f, the second member of this 
equation must be constant for the exterior surface, and for that of the 
shells in equilibrium, and can only vary from one shell to another. 

The two preceding cases reduce to one when the spheroid is homoge 
neous ; for it is indifferent as to the equilibrium whether it is entirely 
fluid, or contains an interior solid nucleus. It is sufficient by No. 556, that 
at the exterior surface we have 

constant = V + a r 2 [Z^+ Z+ r Z + c.}. 

If we substitute in this equation for V its value given by formula (3) of 
No. 555, and if we observe that by No. 556, Y (0) disappears by taking for 
a the radius of a sphere of the same volume as the spheroid, and that 
Y (l is nothing when we fix the origin of coordinates at the center of the 
spheroid; we shall have 

constant = ^l + ^L i {J_ YB , + JL. Y +j 5f.Y+ & c.} 

+ a r 2 [Z !0 > + Z (2 > + r Z + r 2 Z + & c .} 

Substituting in the equation of the surface of the spheroid for r its value 
at the surface 1 + a y, or 

a + a a Y (2) + Y< 3 > + Y -> + &c.} 
which gives 

const. = -^a* 8< 7 a * {5 Y(2) + -f- Y(3) +4 YW + &C l 

+ a a* {ZW + Z^ + a Z + a 2 Z + &c.} 

We shall determine the arbitrary constant of the first member of this 
equation, by means of this equation, 

const. = a 2 + a 8 Z w > ; 



we shall then have by comparing like functions, that is to say, such as are 
subject to the same equation of partial differences, 

R l 



264 A COMMENTARY ON [SECT. XII. & XIII. 



i being greater than unity. The preceding equation may be put under the 
form 



the integral being taken from r = to r = a. The radius a (1 ay) 
of the surface of the spheroid will hence become 




We may put this equation under a finite form, by considering that we 
have by the preceding No. 



so that the integraiy* d r JZ ^ + r Z 3} + &c.} is easily found by known 
methods. 

569. The equation (1) of 567 not only has the advantage of showing the 
figure of the spheroid, but also that of giving by differentiation the law of 
gravity at its surface ; for it is evident that the second member of this 
equation being the integral of the sum of all the forces with which each 
molecule is animated, multiplied by the elements of their respective direc 
tions, we shall have that part of the resultant which acts along the radius 
r, by differentiating the second member relatively to r; thus calling p 
the force by which a molecule of the surface is sollicited towards the center 
of gravity of the spheroid, we shall have 

p = (^) ~ d {r 8 Z<> + r 2 Z + r 3 Z + r 4 Z< + &c.|. 

If we substitute in this equation for (. \, its value at the surface 

2 V 

it a + , given by equation (2) of No. 554, and for V, its value given 
o *& n 

by equation (1) of No. 567; we shall have 



p = * a _ a a {z + a Z (3 > a 2 Z< 4 > 
> 



&c.} (3) 



BOOK L] NEWTON S PRINCIPIA. 265 

r must be changed into a after the differentiations in the second mem 
ber of this equation, which by the preceding No. may always be reduced 
to a finite function. 

p does not represent exactly gravity, but only that part of it which is 
directed towards the center of gravity of the spheroid, by supposing it re 
solved into two forces, one of which is perpendicular to the radius r, and 
the other p is directed along this radius. The first of these two forces is 
evidently a small quantity of the order a ; denoting it therefore by a 7, 
gravity will be equal to Vp 2 -f- 2 7 2 , a quantity which, neglecting the 
terms of the order a 2 , reduces to p. We may thus consider p as express 
ing gravity at the surface of the spheroid, so that the equations (2) and 
(3) of the preceding No. and of this, determine both the figure of ho 
mogeneous spheroids in equilibrium, and the law of gravity at their 
surfaces ; they contain the complete theory of the equilibrium of these 
spheroids, on the supposition that they differ very little from the sphere. 

If the extraneous bodies S, S , &c. are nothing, and therefore the 
spheroid is only sollicited by the attraction of its molecules, and the cen 
trifugal force of its rotatory motion, which is the case of the Earth and 
primary planets with the exception of Saturn, when we only regard the 
permanent state of their figures ; then designating by a p, the ratio of 
the centrifugal force to gravity at the equator, a ratio which is very nearly 

equal to-, the density of the spheroid being taken for unity; we shall 
find, 



the spheroid is then therefore an ellipsoid of revolution, upon which in 
crements of gravity, and decrements of the radii, from the equator to 
the poles, are very nearly proportional to the square of the sine of the 
latitude, m being to quantities of the order a, equal to this sine. 

a, by what precedes, is the radius of a sphere, equal in solidity to the 
spheroid ; gravity at the surface of this sphere will be f v a ; thus we shall 
have the point of the surface of the spheroid, where gravity is the same as 
at the surface of the sphere, by determining m by the equation 

=-_ + f (m i)j 
which gives 



266 A COMMENTARY ON [SECT. XII. & XIII. 

570. The preceding analysis conducts us to the figure of a homoge 
neous fluid mass in equilibrium, without employing other hypotheses than 
that of a figure diifering very little from the sphere : it also shows that 
the elliptic figure which satisfies this equilibrium, is the only figure 
which does it. But as the expansion of the radius of the spheroid into 
a series of the form a [I + a Y ( ) + a Y (1) + &c.} may give rise to some 
difficulties, we proceed to demonstrate directly, and independently of this 
expansion, that the elliptic figure is the only figure of the equilibrium of 
a homogeneous fluid mass endowed with a rotatory motion ; which by con 
firming the results of the preceding analysis, will at the same time serve 
to remove any doubts we may entertain against the generality of this ana 
lysis. 

First suppose the spheroid one of revolution, and that its radius is a 
(1 + a y), y being a function of m, or of the cosine of the angle 6 which this 
radius makes with the axis of revolution. If we call f any straight line 
drawn from the extremity of this radius in the interior of the spheroid ; p 
the complement of the angle which this straight line makes with the plane 
which passes through the radius a ( 1 + ay) and through the axis of revolu 
tion; q the angle made by the projection of f upon this plane and by the 
radius ; finally, if we call V the sum of all the molecules of the spheroid, 
divided by their distances from the molecules placed at the extremity of 
the radius a (1 + a y) ; each molecule being equal to f 2 d f. d p. d q . 
sin. p, we shall have 

V = i/f /2 dp.dq.sin. p, 

f being what f becomes at its quitting the spheroid. We must now de 
termine f in terms of p and q. 

For that purpose, we shall observe that if we call 4 , the value of 6 rela 
tive to this point of exit, and a (1 + ay ), the corresponding radius of the 
spheroid, y being a similar function of cos. 6 or of m that y is of m ; it 
is easily seen that the cosine of the angle formed by the two sti aight lines 
f and a ( 1 + a y) is equal to sin. p . cos. q ; and therefore that in the 
triangle formed by the three straight lines f, a ( 1 + ay) and a ( 1 + a y ) 
we have 

a*(l + ay ) z = i * 2af (l + a y) sin. p . cos. q + a 2 (l +y) 2 . 

This equation gives for f 2 twa values ; but one of them being of the 
order a 2 is nothing when we neglect the quantities of that order; the 
other becomes 

f /2 = 4 a 2 sin. 2 p cos. 2 q (1 + 2 ay) -f- 4 a a 2 (y y) ; 
which gives 



BOOK I.] NEWTON S PRINCIPIA, 267 

V - 2 a*/dp clq sin, p {(1 + 2 ay) sin. 2 p cos. 2 q+ (y y)J. 
It is evident that the integrals must be taken from p = 0, to p = *, and 
from q = * to q = T ; we shall therefore have 

V = f r a - ?r a 2 y -f- 2 a 2 y*d p . d q . y sin. p . 
y being a function of cos. S } we must determine this cosine in a function 
of p and q; we may therefore in this determination neglect the quantities 
of the order a, since y is already multiplied by a ; hence we easily find 

a cos. 6 = (a P sin. p cos. q) cos. 6 + f sin. p . sin. q . sin. ; 
whence we derive, substituting for P its value 2 a sin. p cos. q, 

in = m cos. 2 p sin. 2 p cos. (2 q + 6). 

Here we must observe, relatively to the integral f y d p . d q . sin. p, 
taken relatively to q from 2q = * to 2 q = < that the result would 
be the same, if this integral were taken from 2 q =r to 2 q = 2 0, 
because the values of m , and consequently of y are the same from 2 q = 
9 to 2 q = 6 as from 2 q = r to 2 q = 2 it 6 ; supposing there 
fore 2 q + 6 = q , which gives 

m = m cos. 2 p sin. 2 p cos. q ; 
we shall have 

V = f TT a 2 | cr a 2 y -f a a 2 /y d p d q sin. p ; 
the integrals being taken from p = to p = * and from q = to q = 
2cr. 

Now if we denote by a 2 N the integral of all the forces extrinsic to the 
attraction of the spheroid, and multiplied by the elements of their direc 
tions ; by 568 we shall have in the case of equilibrium 

constant = V + a 2 N, 
and substituting for V its value, we shall have 

const. = a * y a fy d p . d q sin. p N ; 

an equation which is evidently but .the equation of equilibrium of No. 568, 
presented under another form. This equation being linear, it thence results 
that if any number i of radii a (1 + a y), a (1 + a v), and satisfy it; the 

radius a { + (y + v + &c.)} will also satisfy it. 

1 

Suppose that the extraneous forces are reduced to the centrifugal force 
due to the rotation, and call g this force at the distance 1 from the axis of 
rotation; we shall have, by 567, N = g (1 m 9 ) ; the equation of 
equilibrium will therefore be 

const. = | a v y a/y d p d q sin. p 2 g (1 m 8 ). 

Differentiating three times successively, relatively to m, and observing 

that ( ^ = cos. " p, in virtue of the equation 
\d m / 



268 A COMMENTARY ON [SECT. XII. Sc XIII. 

m = m cos. 2 p sin. 2 p cos. q j 



we shall have 



m 



/ d p d q sin. p cos. 
J 



but we have yd p d q sin. p cos. 6 p = ~j we may therefore put the 
preceding equation under this form, 

0=/d p d q sin. p co, p { J () - () } . 

This equation subsists, whatever m may be; but it is evident, that 
amongst all the values between m = 1 and m = 1, there is one which 
we shah 1 designate by h, and which is such that, abstraction being made 

t q 

of the sign, each of the values of (-, ~ 3 \ will not exceed that which is re 
lative to h ; denoting therefore by H, this latter value, we shall have 
= / d p d q sin. p cos. B p { I H - (fjlZl) } . 

1 q / 

The quantity H (-^ *r*\ h as evidently the same sign as H, and 

the factor sin. p . cos. 6 p, is constantly positive in the whole extent of the 
integral; the elements of this integral have, therefore, all of them the 
same sign as H ; whence it follows that the entire integral cannot be no 
thing, at least H cannot be so, which requires that we have generally 

= (-: ZjY whence by integrating we get 

y =r 1+ m. m +n.m 2 ; 
1, m, n, being arbitrary constants. 

If we fix the origin of the radii in the middle of the axis of revolution, 
and take for a the half of this axis, y will be nothing when m = I and 
when m = 1, which gives m = and 11 = 1 ; the value of y thus 
becomes, 1 (1 m 2 ); substituting in the equation of equilibrium, 
const. = | a y ay y d p d q sin. p g (1 m *) ; 

1 "" X 

we shall find a 1 = - ^ = -r a <p, a <p being the ratio of the centrifugal 
16 K 4> 

force to the equatorial gravity, a ratio which is very nearly equal to ^ ; 
the radius of the spheroid will therefore be 
. {1+^(1 -m )}; 

whence it follows that the spheroid is an ellipsoid of revolution, which is 
conformable to what precedes. 



BOOK I.] NEWTON S PRINCIPIA. 269 

Thus we have determined directly and independently of series, the 
figure of a homogeneous spheroid of revolution, which turns round its 
axis, and we have shown that it can only be that of an ellipsoid which 
becomes a sphere when <p = ; so that the sphere is the only figure of 
revolution which would satisfy the equilibrium of an imrnoveable homo 
geneous fluid mass. 

Hence we may conclude generally, that if the fluid mass is sollicited 
by any very small forces, there is only one possible figure of equilibrium . 
or, which comes to the same, there is only one radius a (1 + y) which 
can satisfy the equation of equilibrium, 

const. = a it . y a ,/y d p . d q sin. p N; 

y being a function of 6 and of the longitude &, and y being what y be 
comes when we change Q and into (i and . Suppose, in fact, that 
there are two different rays a (1 + ay) and a(l + y + v) which 
satisfy this equation ; we shall have 

const. = a f (y -j- v) a f(y + v/ ) d p d q sin. p N. 
Taking the preceding equation from this, we shall have 

const. = it v y v d p d q sin. p. 

This equation is evidently that of a homogeneous spheroid in equili 
brium, whose radius is a (1 + a v), and which is not sollicited by any 
force extraneous to the attraction of its molecules. The angle -a disappear 
ing in this equation, the radius a (1 + a v) will still satisfy it if -a be suc 
cessively changed to + d -a, -a -f 2 d &, &c., whence it follows, that if 
we call v 1} v 2 , &c. what v becomes in virtue of these changes; the 
radius 

n 1 + a vdw-f- a vidw + av 2 dw+ &c.}, 
or 

a (1 + a/v d *r), 

will satisfy the preceding equation. If we take the integral fv d -a from 
a = to -o = 2 or, the radius a (1 + aj" \ d &} becomes that of a sphe 
roid of revolution, which, by what precedes, can only be a sphere : see 
the condition which results for v. 

Suppose that a is the shortest distance of the center of gravity of the 
spheroid whose radius is a (I + a v), to the surface, and fix the pole or 
origin of the angle 6 at the extremity of a ; v will be nothing at the pole, 
and positive every where else; it will be the same for the integraiyVd -a. 
But, since the center of gravity of the spheroid whose radius is a (l+v), 
is at the center of the sphere whose radius is a, this point will, in like 
manner, be the center of gravity of the spheroid whose radius is 



S70 A COMMENTARY ON; [SECT. XII. & XIII. 

a (1 -f. ufv d ) ; the different radii drawn from this center to the sur 
face of this last spheroid are therefore unequal to one another, if v is not 
nothing ; there can only therefore be a sphere in the case of v = ; thus we 
learn for a certainty, that a homogeneous spheroid, sollicited by any small 
forces whatever, can only be in equilibrium in one manner. 

571. We have supposed that N is independent of the figure of 
the spheroid; which is what very nearly takes place when the forces, 
extraneous to the action of the fluid molecules, are due to the centri 
fugal force of rotatory motion, and to the attraction of bodies exterior 
to the spheroid. But if we conceive at the center of the spheroid a finite 
force depending on the distance r, its action upon the molecules placed at 
the surface of the fluid, will depend on the nature of this surface, and 
consequently N will depend upon y. This is the case of a homogeneous 
fluid mass which covers a sphere of a density different from that of the 
fluid ; for we may consider this sphere as of the same density as the fluid, 
and .may place at its center a force reciprocal to the square of the dis 
tances; so that, if we call c the radius of the sphere, and fits density, that 
of the fluid being taken for unity, this force at the distance r will be equal 

3 / 1 N 

to * K . Y - Multiplying by the element d r of its direction 

c 3 f p ]) 

the integral of the product will be <n . -, a quantity which we 

must add to a e N ; and since at the surface we have r = a (1 + a y), in 
the equation of equilibrium of the preceding No., we must add to N, 



This equation will become 

4 C6 CT j . C I r f i i XT 

const. - 5 "5 1 + (g 1) . f y /y d p . d q sin. p N. 
-~ o v. a J 

If we denote by a (1 + ay + a v), a new expression of the radius of 
the spheroid in equilibrium, we shall have to determine v, the equation 

f ^ 1 

const. T 1 1 + (s J) r*j / v/ tl P d l l sin - P 5 

an equation which is that of the equilibrium of the spheroid, supposing it 
immoveable, and abstracting every external force. 

If the spheroid is of revolution, v will be a function of cos. 6 or m only; 
but in this case we may determine it by the analysis of the preceding No. ; 
for if we differentiate this equation i + 1 times successively relatively to 
in, we shall have 

= i T -f 1 -f fi- 



_ 

a 3 -d m 



BOOK I.] NEWTON S PR1NCIP1A. 271 

but we have 

/d p d q sin. p cos. 2 + 2 p = ^ .-^ ; 

the preceding equation may therefore be put under this form, 

(2i + 3 
0=/dpdq siii.pcos.* + 8 p _ 



We may take i such that, abstraction being made of the sign, we have 

-Me -.> 



Supposing, therefore, that i is the smallest positive whole number which 
renders this quantity greater than unity, we may see, as in the preceding No., 

/d + 1 v\ 
that this equation cannot be satisfied unless we suppose (-= - njri) 

which gives 

v = m i + Am i - 1 -i-Bm i - 2 + &c. 

Substituting in the preceding equation of equilibrium for v, this value, 

and for v 

m 5 + A m 1 - 1 + Bm i - 2 + Sec. 

m being by the preceding No. equal to m cos. 2 p sin. 2 p cos. q , first 
we shall find 



which supposes g equal to or less than unity ; thus, whenever a, c, and g 
are not such as to satisfy this equation, i being a positive whole number, 
the fluid can be in equilibrium only in one manner. Then we shall have 



so that 



there are, therefore, generally two figures of equilibrium, since a v is sus 
ceptible of two values, one of which is given by the supposition of = 0, 
and the other is given by the supposition of v being equal to the preced 
ing function of m. 

If the spheroid has no rotatory motion, and is not sollicited by any ex 
traneous force, the first of these two figures is a sphere, and the second 
has for its meridian a curve of the order i. These two curves coincide in 
the case of i = J, because the radius a (1 + am) is that of a sphere in 
which the origin of the radii is at the distance a from its center , but then 
it is easy to see that e = 1, that is, the spheroid is homogeneous, a result 
agreeing with that of the preceding No. 



272 A COMMENTARY ON [SECT. XII. & XIII. 

572. When we have figures of revolution which satisfy the equilibrium, 
it is easy to obtain those which are not of revolution by the following 
method. Instead of fixing the origin of the angle 6 at the extremity of 
the axis of revolution, suppose it at the distance 7 from this extremity, and 
call ff the distance from this same extremity of the point of the surface 
whose distance from the new origin of the angle 6 is 6. Call, moreover, 
ta /3 the angle comprised between the two arcs 6 and 7 ; we shall 

have 

cos. (f = cos. 7 cos. 6 + sin. 7 sin. 6 . cos. (w /3) ; 

designating therefore by r . (cos. tf) the function 



the radius of the immoveable spheroid in equilibrium, which we have seen 
is equal to a {1 + " r - ( cos - $ )}> wu *l be 

a + a r. {cos. 7 . cos. 6 + sin. 7 . sin. 6 cos. (& (3)} ; 

and although it is a function of the angle *r, it belongs to a solid of revo 
lution, in which the angle d is not at the extremity of the axis of revo 
lution. 

Since this radius satisfies the equation of equilibrium, whatever may be 
a, /3, and 7, it will also satisfy in changing these quantities into a , /3 , 7 , 
a "> |8", 7") &c. whence it follows that this equation being linear, the radius 
a + a a r . {cos. 7 cos. Q + sin. 7 sin. Q cos. ( /3 )} 
+ a aT . [cos. y cos. 6 -f- sin. y* sin. 6 cos. (^ j3 )] 
+ &c. 

will likewise satisfy it. The spheroid to which this radius belongs is no 
longer one of revolution ; it is formed of a sphere of the radius a, and of 
any number of shells similar to the excess of the spheroid of revolution 
whose radius is a + a a r . (m) above the sphere whose radius is a, these 
shells being placed arbitrarily one over another. 

If we compare the expression of r. (cos. $ ) with that of P (i > of No. 567, 
we shall see that these two functions are similar, and that they differ only 
by the quantities 7 and /3, which in P W are v ai}d ^ and by a factor in 
dependent of m and vr ; we have, therefore, 




d 

It is easy hence to conclude, that if we represent by a Y (i > the function 

a . r . {cos. 7 cos. 6 -j- sin. 7 sin. d . cos. (-a (3 )} 
+ a! . r . {cos. 7 cos. Q + sin. 7 sin. 6 . cos. (v /3 )} 



BOOK I.] NEWTON S PRINCIP1A. 273 

Y (l) will be a rational and entire function of m, VI nT 2 cos. &, 



VI m 2 sin. *, which will satisfy the equation of partial differences, 






choosing for Y 9 therefore, the most general function of that nature, the 
function a (1 + Y (i) ) will be the most general expression of the equili 
brium of an immoveable spheroid. 

We may arrive at the same result by means of the series for V in 555 ; 
for the equation of equilibrium being, by the preceding No., 

const. = V + a 2 N; 
if we suppose that all the forces extraneous to the reciprocal action of the fluid 

molecules, are reducible to a single attractive force equal to f it. C , 

placed at the center of the spheroid, by multiplying this force by the ele 
ment d r of its direction, and then integrating, we shall have 



and since at the surface r = a (1 + y) the preceding equation of equi 
librium will become 



c 3 
const. = V + t . (1 f)y. 

fl 



Substituting in this equation for V its value given by formula (3) of 
No. 555, in which we shall put for r its value a (1 -f a y), and by sub 
stituting for y its value 

Y<> + YW + Y + &c.; 
we shall have 



= 



the constant a being supposed such, that const. = $ ir a 2 . This equation 
gives Y ) = 0, Y ^ = 0, Y = 0, &c. unless the coefficient of one of these 
quantities, of Y W for example, is nothing, which gives 
(I x c 3 __ 2 i 2 

~ s ~ti r ~~ 2i + 1 

i being a positive whole number, and in this case all these quantities ex 
cept Y W are nothing ; we shall therefore have y = Y (i >, which agrees 
with what is found above. 

Thus we see, that the results obtained by the expansion of V into a se- 
VOL. II. S 



274 A COMMENTARY ON [SECT. XII. & XIII. 

ries, have all possible generality, and that no figure of equilibrium has 
escaped the analysis founded upon this expansion ; which confirms what 
we have seen a priori, by the analysis of 555, in which we have proved 
that the form which we have given to the radius of spheroids, is not arbi 
trary but depends upon the nature itself of their attractions. 

573. Let us now resume equation (J) of No. 567. If we therein sub- 
stitutefor V its value given by formula (6) of No. 558, we shall have rela 
tively to the different fluid shells 



/{] TT f 

Hsfcr/f ft*4 4r/f d 



n r 

a W+y 



+ a r 2 JZ<> + Z + r Z + r 2 Z^ + &c.} ; . . . . (1) 
the differentials and integrals being relative to the variable a; the two first 
integrals of the second member of this equation must be taken from a = a to 
a = 1, a being the value of a, relative to the leveled fluid shell, which we are 
considering, and this value at the surface being taken for unity : the two last 
integrals ought to be taken from a = to a = a : finally, the radius r 
ought to be changed into a ( 1 + ay) after all the differentiations and in 
tegrations. In the terms multiplied by a it will suffice to change r into 

a ; but in the term -^- f % d . a 3 we must substitute a (1 + a y) for r ; 
o r 

which changes it into this 

4 ?r 

3 a *.- . 

and consequently, into the following 

_ Y (1 > a Y (3) &C.L fp d a 3 . 



w .* 

Hence if in equation (1) we compare like functions, we shall have 
A C - = 2 * fe d a 2 + 4 a f fp d (a 2 Y^ x - 1 - r ~ A - 3 

J J *> t/ & V 



3 a 



> ;t a 

the two first integrals of the second member of this equation being taken 
from a = a to a = 1, the three other integrals must be taken from a 
= to a = a. This equation determining neither a nor Y (0) , but only a 
relation between them, we see that the value of Y (0) is arbitrary, and may 
be determined at pleasure. We shall have then, i being equal to, 01 
greater than unity, 



BOOK I.] NEWTON S PRINCIPIA. 73 

4 a ^ j /Y Ci) \ 4 cr , ,, , , 

= : fe d. ( r-3 j = Y ]) ft d a 3 

2i+l J s Va - V 3 a J 



the first integral being taken from a a, to a = ], and the two others 
being taken from a = to a = a. This equation will give the value of 
Y (i) relative to each fluid shell, when the law of the densities g shall be 
known. 

To reduce these different integrals within the same limits, let 

4 T 



the integral being taken from a = to a = 1 ; Z (i) will be a quantity in 
dependent of a, and the equation (2) will become 



3/g d (a * + s Y ) 3 a 2 5 + 1 Z (i > ; 
all the integrals being taken from a = to a = a. 

We may make the signs of integration disappear by differentiating re 
latively to a, and we shall have the differential equation of the second 
order, 

/d Yx Ji(j+ 1) 6 g a 1 6ga 2 /d Y * 

\da z ) \ a 2 / f da 3 / /g d. a 3 V d a ) 

The integral of this equation will give the value of Y (l) with two arbi 
trary constants ; these constants are rational and entire functions of the 
order i, of m, VI m 2 . sin. &, and VI m 2 . cos. ^-, such, that re 
presenting them by U (i) , they satisfy the equation of partial differences, 




dm / 1 m 2 

One of these functions will be determined by means of the function 
Z (i) which disappears by differentiation, and it is evident that it will be a 
multiple of this function. As to the other function, if we suppose that 
the fluid covers a solid nucleus, it will be determined by means of the 
equation of the surface of the nucleus, by observing that the value of 
Y :i) relative to the fluid shell contiguous to this surface, is the same as 
that of the surface. Thus the figure of the spheroid depends upon the 
figure of the internal nucleus, and upon the forces which sollicit the 
fluid. 

574. If the mass is cntirebjiKfluid, nothing then determining one of the 
arbitrary constants, it would seem that there ought to be an infinity of 

S 2 



276 A COMMENTARY ON SECT. XII. & XIII. 

figures of equilibrium. Let us examine this case particularly, which is 
the more interesting inasmuch as it appears to have subsisted primi 
tively for the celestial bodies. 

First, we shall observe that the shells of the spheroid ought to decrease 
in density from the center to the surface ; for it is clear that if a denser 
shell were placed above a shell of less density, its molecules would pene 
trate into the other in the same manner that a ponderous body sinks into 
a fluid of less density ; the spheroid will not therefore be in equilibrium. 
But whatever may be its density at the center, it can only be finite ; re 
ducing therefore the expression of g into a series ascending relatively to 
the powers of a, this series will be of the form /3 y . a n &c. |8, y and 
n being positive ; we shall thus have 

3 y . a " & . 



(n + 3) /3 
and the differential equation in Y w will become 




To integrate this equation, suppose that Y (i) is developed into a series 
ascending according to the powers of a, of this form 

Y U) = a s . U + a 5 . U + &c. ; 
the preceding differential equation will give 

i + 2) a s ~ 2 U < + &c. 



= j_ (g+ 1)a .- S . U 0> +(8 > + i) a *-au +&c.S . (e) 
(n + A ) P 

Comparing like powers of a, we have (s + i + 3) (s i + 2) = 0, 
which gives = i 2, and s = i 3. To each of these values of 
s, belongs a particular series, which, being multiplied by an arbitrary, will 
be an integral of the differential equation in Y (i > ; the sum of these two in 
tegrals will be its complete integral. In the present case, the series which 
answers to s = i 3 must be rejected ; for there thence results for a 
Y (i) , an infinite value, when a shall be infinitely small, which would render 
infinite the radii of the shells which are infinitely near to the center. Thus 
of the two particular integrals of the expression of Y w , that which answers 
to s = i 2 ought alone to be admitted. This expression then, contains 
no more than one arbitrary which will be determined by the function Z (i) . 

Z (1 > being nothing by No. 567, Y (1) is likewise nothing, so that the 
center of gravity of each shell, is at the center of gravity of the entire 



BOOK I.] NEWTON ^ PRINC1PIA. 277 

spheroid. In fact the differential equation in Y (i) of the preceding No. 
gives 



/d YWv . /2x 6g 

Vda 2 /"VaV /gd. 



a) _ 6ga 
3 * /gd. 



We satisfy this equation by making Y (1) = - , U (l) being indepen- 



ft 



dent of a. This value of Y (1) is that which answers to the equation 
s = { 2 ; it is, consequently, the only one which we ought to admit. 
Substituting it in the equation (2) of the preceding No., and supposing 
Z (1) = 0, the function U (1) disappears, and consequently remains arbitrary; 
but the condition that the origin of the radius r is at the center of gravity 
of the terrestrial spheroid, renders it nothing ; for we shall see in the follow 
ing No. that then Y (1) is nothing at the surface of every spheroid covered 
over with a shell of fluid in equilibrium ; we shall have, therefore, in the 
present ease U (1) = ; thus, Y (1) is nothing relatively to all the fluid shells 
which form the spheroid. 

Now consider the general equation, 

Y = a s . U + a s/ . U + &c. ; 

s being, as we have seen, equal to i 2, s is nothing or positive, when i 
is equal to or greater than 2; moreover, the functions U w , U //(l) , &c. are 
given in U (i) , by the equation (e) of this No. ; so that we have 

Y = h. U (i >; 

h being a function of a, and U ( being independent of it. If we substi 
tute this value of Y in the differential equation in Y l , we shall have 
d 2 h f 6g a 3 ) _h_ 6 ga 2 dji 

"cl"a^" =: V (l 1) ""77d7a"= r j a 2 "" /gd.a 3 da 

The product i (i + 1) is greater than -7* 4 - r> when i is equal to or 

t/ fa 

e a 3 
greater than 2, for the fraction ,. g - - is less than unity : in fact its 

J S d a 
denominator f d . a 3 is equal to a 3 f a 3 d g, and the quantity 

fa 3 d g is positive, since g decreases from the center to the surface. 
Hence it follows that h and -r - are constantly positive, from the 

center to the surface. To show this, suppose that both these quantities are 
positive in going from the center; d h ought to become negative before h, 
and it is clear that in order to do this it must pass through zero ; but 
from the instant it is nothing, d 2 h becomes positive in virtue of the pre 
ceding equation, and consequently d h begins to increase ; it can never 
therefore become negative. Whence it follows that h and d h always pre- 

S3 



278 A COMMENTARY ON [SECT. XII. & XIII. 

serve the same sign from the center to the surface. Now both of these 
quantities are positive in going from the center ; for we have in virtue of 
equation (e), s 2=s + n 2, which gives s = i + n 2 ; hence 
we have 



s + i + 3) ( S - i + a) u- B = 6 n 5 + 



Ufl> 



V n + 3) p 

whence we derive 

U(i) _ 6(i-l)y.UM 

(n + 8)(2i + n + l)./3 
we shall therefore get 

6 (i 1) y. a 1 *"- 2 
h = al - 3 +(n + 3)(2i + n 

dh 6i-i 



i SU - 8 . - -. 

d a - 1 (n+3)(2i + n+l)/3 , + 

7, /?, n, being positive, we see that at the center h and d h are positive, 
when i is equal to or greater than 2 ; they are therefore constantly positive 
from the center to the surface. 

Relatively to the Earth, to the Moon, to Jupiter, &c. Z (i > is nothing or 
insensible, when i is equal to or greater than 3 ; the equation (2) of the 
preceding No. then becomes 

0= ^3a^ 

the first integral being taken from a = a, to a = 1, and the two others 
being taken from a = 0, to a = a. At the surface where a = 1, this equa 
tion becomes 

= { (2 i+ 1) h/d. a 3 + 3/d (a^h)}. U; 
an equation which we can put under this form 

= J_(2i 2)gh + (2i+l) h/a 3 dg 3/a i + Mi.d^ U . 

d g is negative from the center to the surface, and h increases in the 
same interval; the function (2 i + 1) \\f a 3 d g 3y"a + 3 h d g is therefore 
negative in the same interval ; thus in the preceding equation the coeffi 
cient of U (i) is negative and cannot be nothing at the surface ; U (i) ought 
therefore to be nothing, which gives Y w = ; the expression of the ra 
dius of the spheroid thus reduces to a + a a {Y (0) + Y (2) ] ; that is to say, 
that the surface of each leveled shell of the spheroid is elliptic, and conse 
quently its exterior surface is elliptic. 

Z (2 >, relatively to the Earth is, by No. 567, equal to - (m 2 ) ; 

& X 

the equation (2) of the preceding No. gives therefore 



BOOK I.] NEWTON S PR1NCIPIA. 279 

0= &r 

At the surface, the first integral^ d h is nothing; we have therefore at 
this surface where a = 1, 



|J(8) = 



Let a p, be the ratio of the centrifugal force to the equatorial gravity ; 
the expression of gravity to quantities of the order a, being equal to 
I ""ft d . a 3 ; we shall have g = f ir a ipfg d . a 3 ; wherefore 



U (2 > = 



2 / g .d(a*b) ; 



2h = 

5 j g . a a a 

comprising therefore in the arbitrary constant a, what we have taken for 
unity, the function 

a h <p 



5 * f i .a 2 d a 

the radius of the terrestrial spheroid at the surface will be 

a h p (1 m 2 ) 

*! O / _ ,1 / 5 K \ * 



5 y"g.a 2 d a 

The figure of the earth supposed fluid, can therefore only be that of an 
ellipsoid of revolution ; all of whose shells of constant density are elliptic, 
and of revolution, and in which, the ellipticities increase, and the densities 
decrease from the center to the surface. The relation between the ellip 
ticities and densities is given by the differential equation of the second 
order, 

d h _ 6_h / ga 3 \ 2ga 2 dji 

da 2 " a 2 X. ~ 3/ a 2 d a/ ~fg . a " d a cTa 

This equation is not integrable by known methods except in some par 
ticular suppositions of the densities g ; but if the law of the ellipticities 
were given, we should easily obtain that of the corresponding densities. 
We have seen that the expression of h given by the integral of this equa 
tion contains, in the present question, only one arbitrary, which disappears 
from the preceding value of the radius of the spheroid ; there is therefore 
only one figure of equilibrium differing but little from a sphere, which is 
possible, and it is easy to see that the limits of the flattening of this figure 

are ^ and <p, the former of which corresponds to the case where all 



280 A COMMENTARY ON [SECT. XII. &XI11. 

the mass of the spheroid is collected at its center, and the second to the 
case where this mass is homogeneous. 

The directions of gravity from any point of the surface to the center do 
not form a straight line, but a curve whose elements are perpendicular to 
the leveled shells which they traverse : this curve is the orthogonal tra 
jectory of all the ellipses which by their revolution form these shells. To 
determine its nature, take for the axis, the radius drawn from the center 
to a point of the surface, d being the angle which this radius forms with 
the axis of revolution. We have just seen that the general expression of 
any shell of the spheroid is a + a k . a h . (1 m 2 ), k being independent 
of a : whence it is easy to conclude that if we call a y , the ordinate let 
fall from any point of the curve upon its axis, we shall have 

/ i ( r h d a) 

ay = a a k . sin. 2 6 -\ c / Y , 

c being the entire value of the integral /* - , taken from the center to 

the surface. 

575. Now consider the general case in which the spheroid always fluid 
at its surface, may contain a solid nucleus of any figure whatever, but dif 
fering but little from the sphere. The radius drawn from the center of 
gravity of the spheroid to its surface, and the law of gravity at this sur 
face have some general properties, which it is the more essential to con 
sider, inasmuch as these properties are independent of every hypothesis. 

The first of these properties is, that in the state of equilibrium the 
fluid part of the spheroid must always be disposed so, that the function 
Y (1) may disappear from the expression of the radius drawn from the cen 
ter of gravity of the whole spheroid to its surface ; so that the center of 
gravity of this surface coincides with that of the spheroid. 

To show this, we shall observe that R being supposed to represent the 
radius drawn from the center of gravity of the spheroid to any one of its 
molecules, the expression of this molecule will be g R 2 . d R . d m . d , 
and we shall have by 556, in virtue of the properties of the center of 

gravity, 

=/g R 3 . dR.dm.d^.m; 

= R 3 . d R . d m . d w . V 1 m 2 . sin. w; 



=/g R 3 . d R.dm.dw. V 1 m ". cos. . 

Conceive the integral f g R 3 . d R taken relatively to R from the origin 
of R to the surface of the spheroid, and then developed into a series of 
the form 

&c. ; 



BOOK I.] NEWTON S PRINCIPIA. 

N (i) being whatever i may be, subject to the equation of partial differ 



ences 




d m 
we shall have by No. 556, when i is different from unity, 



= /N. m d m . d * ; =/N w . d m . d . V I in 2 . sin. 
and 



0=/N u >.dm.dw. VI m 2 . cos. *. 

The three preceding equations given by the nature of the center of 
gravity, will become 



=/N< 1 >mdm.dj =/N^dm.d w . V 1 m 2 .sin.*r; 
=/N (1 > d m . d * . V 1 m 2 . cos. 9 . 
N U) is of the form 



H m + H . V I m 2 . sin. * + H". V 1 m 2 . cos. . 
Substituting this value, in these three equations, we shall have 

H = 0; H 7 = 0; H" = 0; 

where N (1) = ; this is the condition necessary that the origin of II is at 
the center of gravity of the spheroid. 

Now let us see, what N (1) becomes relatively to the spheroids differing 
little from the sphere, and covered over with a fluid in equilibrium. In 
this case we have R = a (1 + a y), and the integral fg. R 3 . d R, be 
comes ./ d . [a* (1 + 4 a y)}, the differential and integral being rela 
tive to the variable a, of which g is a function. Substituting for y its va 
lue Y<> + Y + Y + &c., we shall have 
N< = a/gd (a 4 Y>). 

The equation (2) of No. 573 gives, at the surface where a = 1, and 
observing that Z U) is nothing 

/fd(a*Y<) = Y Vfd.a , 

the value of Y (1) in the second member of this equation, being relative to 
the surface ; thus, N (1) being nothing, when the origin of R is at the cen 
ter of gravity of the spheroid, we have in like manner Y ^ =r 0. 

576. The permanent state of equilibrium of the celestial bodies, makes 
known also some properties of their radii. If the planets did not turn ex 
actly, or at least if they turned not nearly, round one of their three principal 
axes of rotation, there would result in the position of their axes of rota 
tion, changes which for the earth above all would be sensible ; and since 
the most exact observations have not led to the discovery of any, we may 
conclude that long since, all the parts of the celestial bodies, and princi- 



282 A COMMENTARY ON [SECT. XII. & XIII. 

pally the fluid parts of their surfaces, are so disposed as to render stable 
their state of equilibrium, and consequently their axes of rotation. It is 
in fact very natural to suppose that after a great number of oscillations, 
they must settle in this state, in virtue of the resistances which they suffer. 
Let us see, however, the conditions which thence result in the expression 
of the radii of the celestial bodies. 

If we name x, y, z the rectangular coordinates of a molecule d M of 
the spheroid, referred to three principal axes, the axis of x being the axis 
of rotation of the spheroid ; by the properties of these axes as shown in 
dynamics, we have 

0=/xy.dM; 0=/xz.dM; 0=/yz.dM; 
the integrals ought to be extended to the entire mass of the spheroid, 
R being the radius drawn from the origin of coordinates to the molecule 
d M ; 6 being the angle formed by R and by the axis of rotation ; and 
a being the angle which the plane formed by this axis and by R, makes 
with the plane formed by this axis and by that of the principal axes, which 
is the axis of y ; we shall have 

x=Rm; y = R V 1 m z . cos. ^ ; z = R V 1 m 2 . sin. -a ; 

dM = gR*d Rdm.d^. 

The three equations given by the nature of the principal axes of rota 
tion, will thus become 



= J g .R 4 . dR.dm.dar.m V 1 m 2 . cos. -a ; 



=fs . R 4 . dR.dm.d^.m VI m 2 . sin. ; 

=/g.R 4 . d R.dm.d .(! m 2 ) sin. 2*. 

Conceive the integral fg R 4 d R taken relatively to R, from R = 0, 
to the value of R at the surface of the spheroid, and developed into a 
series of the form U (0 > + U (1) + U (2) + &c. ; U (i) being, whatever i may 
be, subject to the equation of partial differences, 



m. , 



We shall have by the theorem of No. 556, where i is different from 2, 
and by observing that the functions m V 1 m z . cos. #, m V 1 m 2 . sin. , 
and (1 m 2 ) sin. 2 v, are comprised in the form Uff- ; 

= / U (l) . d m . d -a . m . VI m z . cos. r ; 
= /U W. d m . d . m . V 1 m 2 . sin. ; 
=/U (i >. dm.d .(! m 2 ) sin. 2 . 



BOOK I.] NEWTON S PRINCIPIA. 283 

The three equations relative to the nature of the axes of rotation, will 
thus become 

=r / U (8 >. d m . d * . m . VI m 2 . cos. v ; 
= /U W. d m . d . m . V 1 m 2 . sin. * ; 
= /U. dm.dw. (1 m 2 ) sin. 2 w . 

These equations therefore depend only on the value of U (2) : this value 
is of the form 

H (m 2 I) + H m V 1 m 2 . sin. + H" m V 1 m 2 . cos. + 

H " (1 m 2 ) sin. 2 * + H"" (1 m 2 ) cos. 2 : 
substituting it in the three preceding equations, we shall have 

H = 0; H" = 0; H " = 0. 

It is to these three conditions that the conditions necessary to make the 
three axes of x, y, z the true axes of rotation are reduced, and then U (2) 
will be of the form 

H (m 2 i) + H"" (1 m 2 ) cos. 2 ~. 

When the spheroid is a solid differing but little from the sphere, and 
covered with a fluid in equilibrium, we have R = a (1 + y), and con 
sequently 

ft R 4 . d R = /g d. {a 5 . (1+ 5 a y)}. 

If we substitute for y, its value Y <> + Y (1) + Y ( ~> + &c. ; we shall 
have 

U = a/^d (a 5 Y (2 >). 
The equation (2) of No. 573, gives for the surface of the spheroid, 

~f s d (a 5 Y) = | * YW/f d - a 3 Z^ 2 ; 

Y w and Z (2) in the second member of this equation being relative to the 
surface ; we have therefore, 

U = f a YCygd.a 3 5aZ(8) . 
The value of Z 2 ) is of the form 

-f- ( m 2 i) + g m V 1 m 2 . sin. * + g" 

+ g" (l m s )sin.2 w + g r/// (l m 2 ) cos. 
and that of Y is of the form 



m Vn 2 . C os. 



h (m 2 i) + h m V 1 in 2 , sin. * + h" m V 1 m 2 . cos. 
+ li /x/ . ( 1 m 2 ) sin. 2 * + h w/ (1 m 2 ) cos. 2 *r. 
Substituting in the preceding equation, these values, and H (m 2 _ 
+ H"" (1 m 2 ) cos 2 w , for U ; we shall have 



284 A COMMENTARY ON [SECT. XII. & XIII. 

Such are the conditions which result from the supposition that the sphe 
roid turns round one of its principal axes of rotation. This supposition 
determines the constants h , h", h" by means of the values g , g", g" ; 
but it leaves indeterminate the quantities h and h x/// as also the functions 
Y< 3 >, Y<, &c. 

If the forces extraneous to the attraction of the molecules of the sphe 
roid are reduced to the centrifugal force due to its rotatory motion ; we 
shall have g = 0, g" = 0, g" = ; wherefore h = 0, h" = 0, h" = 0, 
and the expression of Y l 2) , will be of the form 

h (m 2 J) + h"" (1 m 2 ) cos. 2 v . 

577. Let us consider the expression of gravity at the surface of the 
spheroid. Call p this force ; it is easy to see by No. 569, that we shall 
have its value by differentiating the second member of the equation (1) of 
573 relatively to r, and by dividing its differential by d r ; which gives 
at the surface 



r [2Z> + 2Z + 3r.Z + 4r*. Z< 4 > + &c.} ; 
these integrals being taken from a = 0, to a = 1. The radius r at the 
surface is equal to 1 -f a y, or equal to 

1 + Y (0) + YW 4. Y + &c.}; 
we shall hence obtain 



P = ~---~ 

+ 4<r/gd. {a 3 Y<> + ^Y) + ?^-Y (2) + &c.} 

o O 

a {2 Z + 2 Z fa > + 3 Z + 4 Z W + &c. j. 

The integrals of this expression may be made to disappear by means of 
equation (2) of No. 573, which becomes at the surface, 

a 1 * 3 Y = *Y .*. a - Z O. 



supposing therefore 

P=f<r/gd.a 3 

o 

we shall have 

p = P + aP. {Y< 8 > + 

a 5 Z + 7 Z < 3 > + 9 Z w + . . . + (2 i+ 1) Z (i > + &c.}. 

By observations of the lengths of the seconds pendulum, has been re 

cognised the variation of gravity at the surface of the earth. By dy 

namics it appears that these lengths are proportional to gravity ; let 



BOOK L] NEWTON S PRINCIPIA. 285 

therefore 1, L be the lengths of the pendulum corresponding to the gravi 
ties p, P ; the preceding equation will give 



Relatively to the earth a Z (2) reduces by 567, to -^ (m 2 i), or, 



which comes to the same, to -- -^. P. (m 2 ^), a <p being the ratio of 



the centrifugal force to the equatorial gravity; moreover, Z (3) , Z w , &c. 
are nothing ; we have therefore 

1 = L + a L. JYW + 2 Y( 3 > + 3 Y< 4 > + . . . + (i 1) YJ 



The radius of curvature of the meridian of a spheroid which has for its 
radius 1 + a y, is 



l + .(lHLZ) + ._ - -.n 

\ d m / \ dm / 

designating therefore by c, the magnitude of the degree of a circle whose 
radius is what we have taken for unity ; the expression of the degree of 




raus s wat we ave taen o 
the spheroid s meridian, will be 





dm 

y is equal to Y^ + Y^ + Y (2 > + &c. We may cause Y^ to disap 
pear, by comprising it in the arbitrary constant which we have taken for 
the unit ; and Y < l > by fixing the origin of the radius at the center of gravity 
of the entire spheroid. This radius thus becomes, 

1 + a jy< 2 > + Y< 3 > + Y< 4 > + &c.}. 
If we then observe that 



the expression of the degree of the meridian will become 



c 



f /d Y^x , /d Y< 3 \ , \ 

acm<(- -) + ( -3 --- ) + &C. f 
(_ \ d m / v d m / 



a c. ; 

1 m 



286 A COMMENTARY ON [SECT. XII. & XIII. 

If we compare these expressions of the terrestrial radius with the length 
of the pendulum, and the magnitude of the degree of the meridian, we 
see that the term a Y (i) of the expression of the radius is multiplied by 
i 1, in the expression of the length of the pendulum, and by i 2 + i 1 
in that of the degree ; whence it follows, that whilst i 1 is considerable, 
this term will be more sensible in the observations of the length of the 
pendulum than in that of the horizontal parallax of the moon which is 
proportional to the terrestrial radius ; it will be still more sensible in the 
measures of degrees than in the lengths of the pendulum. The reason of 
it is, that the terms of the expression of the terrestrial radius undergo two 
variations in the expression of the degree of the meridian ; and each dif 
ferentiation multiplies these terms by the corresponding exponent of m, 
and this renders them the more considerable. In the expression of the 
variation of two consecutive degrees of the meridian, the terms of the ter 
restrial radius undergo three consecutive differentiations; those which 
disturb the figure of the earth from that of an ellipsoid, may thence be 
come very sensible, and the ellipticity obtained by this variation may be 
very different from that which the observed lengths of the pendulum give. 
These three expressions have the advantage of being independent of the 
interior constitution of the earth, that is to say, of the figure and density 
of its shells; so that if we are going to determine the functions Y (2) , Y (3) , 
&c. by measures of degrees of meridians and parallaxes, we shall have 
immediately the length of the pendulum ; we may therefore thus ascertain 
whether the law of universal gravity accords with the figure of the earth, 
and with the observed variations of gravity at its surface. These remark 
able relations between the expressions of the degrees of the meridian and 
of the lengths of the pendulum may also serve to verify the hypotheses 
proper to represent the measures of degrees of this meridian : this will be 
perceptible from the application we now proceed to make to the hypothe 
sis proposed by Bouguer, to represent the degrees measured northward 
in France and at the equator. 

Suppose that the expression of the terrestrial radius is 1 + Y (2) + 
a. Y (4) , and that we have 

= _ B m 4 _ m + 



it is easy to see that these functions of in satisfy the equations of partial 
differences which Y (2) and Y (4 > ought to satisfy. The variation of the de 
grees of the meridian will be, by what precedes, 

{3 A ~ 2 B} 2 + 15acB.m 4 . 



a c 



BOOK I.] NEWTON S PRINCIPIA. 287 

Bouguer supposes this variation proportional to the fourth power of the 
sine of the latitude, or, which nearly comes to the same, to m 4 ; the term 
multiplied by m 2 , therefore, being made to disappear from the preceding 
function, we shall have 

B = !.A; 

thus in this case the radius drawn from the center of gravity of the earth 
at its surface, will be in taking that of the equator for unity, 

7 a A . 

1 3T~ ( 4m +m 4 ). 

The expression of the length 1 of the pendulum, will become, denoting 
by L, its value at the equator, 

L + f f . L m 2 %^ L (16 m 2 + 21 m 4 ). 

o4i 

Finally, the expression of the degree of the meridian, will be, calling c 
its length at the equator, 

105 
c + . A . c . m 4 . 

We shall observe here, that agreeably to what we have just said, the 
term multiplied by m 4 is three times more sensible in the expression of 
the length of the pendulum than in that of the terrestrial radius, and five 
times more sensible in the expression of the length of a degree, than in 
that of the length of the pendulum ; finally, upon the mean parallel it 
would be four times more sensible in the expression of the variation of 
consecutive degrees, than in that of the same degree. According to Bou- 

959 
guer, the difference of the degrees at the pole and equator is ; it is 

Ot) i Do 

the ratio which, on his hypothesis, the measures of degrees at Pello, Paris 

105 

and the equator, require. This ratio is equal to -=-.- . a A ; we have 

<34 

therefore 

a A = 0. 0054717. 

Taking for unity the length of the pendulum at the equator, the va 
riation of this length, in any place whatever, will be 

0. 0054717 

-. [IG m s + 21 m 4 } + f a <p . m . 

By No. 563, we have a p = 0. 00345113, which gives f a p = 0. 0086278, 
and the preceding formula becomes 

0. 0060529. m 2 0. 0033796. m 4 . 



288 A COMMENTARY ON [SECT. XII. & XIII. 

At Pello, where m = sin. 74. 22 , this formula gives 0.0027016 for 
the variation of the length of the pendulum. According to the observa 
tions, this variation is 0.0044625, and consequently much greater; thus, 
since the hypothesis of Bouguer cannot be reconciled with the observations 
made on the length of the pendulum, it is inadmissible. 

578. Let us apply the general results which we have just found, to the 
case where the spheroid is not sollicited by any extraneous forces, and 
where it is composed of elliptic shells, whose center is at the center of 
gravity of the spheroid. We have seen that this case is that of the earth 
supposed to be originally fluid : it is also that of the earth in the hypo 
thesis where the figures of the shells are similar. In fact, the equation 
(2) of No. 573 becomes at the surface where a 1, 



The shells being supposed similar, the value of Y (i) is, for each of 
them, the same as at the surface ; it is consequently independent of a, and 
we have 



When i is equal to or greater than 3, Z ;i) is nothing relatively to the 

i + 3 

earth; besides the factor 1 . . a is always positive ; therefore Y ^ 

is then nothing. Y (1) is also nothing by No. 575, when we fix the origin 
of the radii at the center of gravity of the spheroid. Finally, by No. 577, 
we have Z (2) equal to 



a da ; 
we have therefore 



fg&*d a (1 a 2 ) 
Thus the earth is then an ellipsoid of revolution. Let us consider there 
fore generally the case where the figure of the earth is elliptic and of re 
volution. 

In this case, by fixing the origin of terrestrial radii at the center of 
gravity of the earth, we have 

Y (D - 0; Y< 3 > = 0; Y = ; &c. 



BOOK I.] NEWTON S PR1NCIPIA. 289 

h being a function of a ; moreover we have 

Z (.) _ . Z (3) = ; Z = 0; &C. 



the equation (2) of No. 573 will therefore give at the surface 

= 6./ f d(a s h) + 5. (p-2h)/ed.a 3 . . . (1) 

This equation contains the law which ought to exist to sustain the 
equilibrium between the densities of the shells of the spheroid and their 
ellipticities ; for the radius of a shell being a [I +a Y (0) a h (p 2 |)} 5 
if we suppose, as we may, that Y (0) = ^ h, this radius becomes 
a (1 a h . ,u, 2 ), and a h is the ellipticity of the shell. 

At the surface, the radius is 1 a h . ^ 2 ; whence we see that the de 
crements of the radii, from the equator to the poles, are proportional to 
/* 2 , and consequently to the square of the sines of the latitude. 

The increment of the degrees of the meridian from the equator to the 
poles is, by the preceding No., equal to 3 a h c . ^ 2 , c being the degree 
of the equator ; it is therefore also proportional to the square of the sine 
of the latitude. 

The equation (1) shows us that the densities being supposed to decrease 
from the center to the surface, the ellipticity of the spheroid is less than 
in the case of homogeneity, at least whilst the ellipticities do not increase 
from the surface to the center in a greater ratio than the inverse ratio of 

the square of the distances to this center. In fact, if we suppose h = 2 , 
we shall have 



If the ellipticities increase in a less ratio than ^ , u increases from tlte 

center to the surface, and consequently d u is positive ; besides, d g is ne 
gative by the supposition that the densities decrease from the center to the 
surface; thus 5< /( d uya 3 d g) is a negative quantity, and making at the 
surface 

/fd(aMi) = (h-f)/gd.a 3 , 
f will be a positive quantity. Hence equation (1) will give 

5 <f> 6 f 
~JT 

a h will therefore be less than - - , and consequently it will be less than 

VOL. II. T 



2<)0 A COMMENTARY ON [SECT. XII. & XIII. 

in the case of homogeneity, where d g being equal to nothing f is also equal 
to zero. 

Hence it follows, that in the most probable hypotheses, the flattening oi 

the spheroid is less than y-r ; for it is natural to suppose that the shells 

~r 

of the spheroid are denser towards the center, and that the ellipticities 

increase from the surface to the center in a less ratio than - z , this ratio 

a 

giving an infinite radius for shells infinitely near to the center, which is 
absurd. These suppositions are the more probable, inasmuch as they 
become necessary in the case where the fluid is originally fluid ; then the 
denser shells are, as we have seen, the nearer to the center, and the ellip 
ticities so far from increasing from the surface to the center, on the con 
trary, decrease. 

If we suppose that the spheroid is an ellipsoid of revolution, covered 
with a homogeneous fluid mass of any depth whatever, by calling a the 
semi-minor axis of the solid ellipsoid, and a h its ellipticity, we shall have 
at the surface of the fluid, 

ft d (a 5 h) = h a" h +fe d (a 5 h)j 

the integral of the second member of this equation being taken relatively 
to the interior ellipsoid, from its center to its surface, and the density of 
the fluid which covers it being taken for unity. The equation (1) will 
give for the expression of the ellipticity h, of the terrestrial spheroid, 
_ 5ap jl a /3 +/gda 3 j Gah . a /5 + 6a/gd(a 5 h) . 
410 a 3 + 10./gd.a 3 






the integrals being taken from a = to a 

Let us now consider the law of gravity, or which comes to the same, 
that of the length of the pendulum at the elliptic surface in equilibrium. 
The value of 1, found in the preceding No., becomes in this case 

1 = L + L J| <f> hj (m 8 ) ; 

making, therefore, L = L i a L (f p h), we shall have, in neglecting 
quantities of the order a \ 

1 = L + L (|f h)//, 2 ; 

an equation from which it results that L is the length of the seconds 
pendulum at the equator, and that this length increases from the equator 
to the poles, proportionally to the square of the sine of the latitude. 

If we call a t the excess of the length of the pendulum at the pole above 
its length at the equator, divided by the latter, we shall have 
a t a (f <p h); 



BOOK I.] NEWTON S PRINCIPIA. 291 

and consequently 

ae + ah = ->ap; 

a remarkable equation between the ellipticity of the earth and the varia 
tion of the length of the pendulum from the equator to the poles. In the 
case of homogeneity ah = f a <p ; hence in this case a s = ah; but if 
the spheroid is heterogeneous, as much as a h is above or below ^ a <p } so 
much is a s above or below the same quantity. 

579. The planets being supposed covered with a fluid in equilibrium, it 
is necessary, in the estimate of their attractions, to know the attraction of 
spheroids whose surface is fluid and in equilibrium : we may express it 
very simply in this way. Resume the equation (5) of No. 558 ; the signs 
of integration may be made to disappear by means of equation (2) of No. 
573, which gives at the surface of the spheroid, 



thus fixing the origin of the radii r at the center of gravity of the spheroid 
which makes Y (^disappear; then observing that Z (1) is nothing, and that Y (0) 

being arbitrary, we may suppose -. Y^ Z (0) = 0, the equation (5) 

9 

of 558, will give 



an expression in which we ought to observe that f% d . a 3 expresses the 

o 

mass of the spheroid, since, in the case of r being infinite, the value of V 
is equal to the mass of the spheroid divided by r. Hence the attraction 

of the spheroid parallel to r will be (-r-) 5 the attraction perpendicu 
lar to this radius, in the plane of the meridian will be -- 

T \ ; finally, the attraction perpendicular to this same radius in the 
direction of the parallel will be 



r V 1 -m 2 
The expression of V, relatively to the earth supposed elliptic, becomes 



M being the mass of the earth. 

T2 



292 A COMMENTARY ON [SECT. XII. & XIII. 

580. Although the law of attraction in the inverse ratio of the square 
of the distance is the only one that interests us, yet equation (1) of 554 
affords a determination so simple of the gravity at the surface of homoge 
neous spheroids in equilibrium, whatever is the exponent of the power of 
the distance to which the attraction is proportional, that we cannot here 
omit it. The attraction being as any power n of the distance, if we de 
note by d m a molecule of the spheroid, and by f its distance from the 
point attracted, the action of d m upon this point multiplied by the element 

d f of its direction, will be d ^ f n . d f. The integral of this quantity, 

d ^ f n + i 
taken relatively to f, is -- , and the sum of these integrals ex 

tended to the entire spheroid is -- - - ; supposing, as in 554, that V = 

/f n + l d ft. 

If the spheroid be fluid, homogeneous, and endowed with rotatory mo 
tion, and not sollicited by any extraneous force, we shall have at the sur 
face, in the case of equilibrium, by No. 567, 

const. = jJLj + $ g r (1 m *), 

r being the radius drawn from the center of gravity of the spheroid at its 
surface, and g the centrifugal -force at the distance 1 from the axis of ro 
tation. 

The gravity p at the surface of the spheroid is equal to the differential 
of the second member of this equation taken relatively to r, and divided 
by d r, which gives 

1 /d Vx 

P = ; f Lr3-*J ff r (1 m s ). 
n + 1 \drJ 

Let us now resume equation (1) of 554, which is relative to the sur 
face, 



= 



2a 2 a 

this equation, combined with the preceding ones, gives 

p = const. + { (n + a 1)r -l} gr(l-m ). 

At the surface, r is very nearly equal to a ; by making them entirely so, 
for the sake of simplicity, we shall have 

p = const. + "~ g (1 m 2 ) 
Let P be the gravity at the equator of the spheroid, and p 



BOOK I.] NEWTON S PRINC1PJA. 293 

the ratio of the centrifugal force to gravity at the equator; we shall 
have 



p = 



whence it follows that, from the equator to the poles, gravity varies as the 
square of the sine of the latitude. In the case of nature, where n = 2, 

we have 

p = P [I + | ap.m*} , 

which agrees with what we have before found. 

But it is remarkable that if n =; 3, we have p = P, that is to say, that 
if the attraction varies as the cube of the distance, the gravity at the sur 
face of homogeneous spheroids is every where the same, whatever may be 
the motion of rotation. 

581. We have only retained, in the research of the figure of the celestial 
bodies, quantities of the order a ; but it is easy, by the preceding analysis, 
to extend the approximations to quantities of the order 2 , arid to superior 
orders. For that purpose, consider the figure of a homogeneous fluid 
mass in equilibrium, covering a spheroid differing but little from a sphere, 
and endowed with a rotatory motion ; which is the case of the earth and 
planets. The condition of equilibrium at the surface gives, by No. 557, 
the equation 

const. = V -- r * (m 2 ). 

i) 

The value of V is composed, 1st, of the attraction of the spheroid co 
vered by the fluid upon the molecule of the surface, determined by the 
coordinates r, 6 9 and w, 2dly, of the attraction of the fluid mass upon this 
molecule. But the sum of these two attractions is the same as the sum of 
the attractions, 1st, of a spheroid supposing the density of each of its shells 
diminished by the density of the fluid; 2dly, of a spheroid of the same density 
as the fluid, and whose exterior surface is the same as that of the fluid. 
Let V be the first of these attractions and V" the second, so that 
V = V +V"; we shall have, supposing g of the order a and equal to g , 

const. = V + V" "-J- . r 2 . (m 2 ). 

/it 

\Ve have seen in 553 that V may be developed into a series of the form 

UW UCD U(2) 

r ~ " T r 

U (i) being subject to the equation of partial differences, 





0= 



= - ___ 

dm 1 m 

T3 



294 A COMMENTARY ON [SECT. XII. & XIII. 

and by the analysis of 561, we may determine U (i) , with all the accuracy 
that may be wished for, when the figure of the spheroid is known. 
In like manner V" may be developed into a series of the form 



U, (i) being subject to the same equation of partial differences as U (i) . If 
we take for the unit of density that of the fluid, we have, by 561, 

U ( 4 * 7 (:) . 

U - (i + 3) (2 i + 1 

r i + 3 b e i n g supposed developed into the series 

ZW + ZW + z< 2 > +&c. 

in which Z (i) is subject to the same equation of partial differences, as U (l) . 
The equation of equilibrium will therefore become 



_ 
i being equal to greater than unity. 

If the distance r from the molecule attracted to the center of the sphe 
roid were infinite, V would be equal to the sum of the masses of the sphe 
roid and fluid divided by r ; calling, therefore, m this mass, we have 
U() -f U/ 0) = m. Carrying the approximation only to quantities of the 
order a 2 , we may suppose 

r = 1.+ a y + a 8 y ; 
which gives 



Suppose 

y = Y (1 > + Y ( -> + Y + &c. 

y / _ Y d) + Y ^ + Y + &c. 

y" = M^ + M> + M + &c. 

Y & t Y x (i) , and M (i) being subject to the same equation of partial differ 
ences as U (i) ; we shall have 



1 

Then observe that U (i) is a quantity of the order a, since it would be 
nothing if the spheroid were a sphere ; thus carrying the approximation 
only to terms of the order a 2 , U will be of this form a U & + 2 U (i) . 
Substituting therefore these values in the preceding equation of equili 
brium, and there changing r into 1 + a y + 2 y , we shall have to quan 
tities of the order 3 , 



BOOK I.] NEWTON S PRINCIPIA. 295 

const. = fA [I ay + 2 y 2 2 y } 

"a U (i) + 2 U" W (i + 1) a* y U 



+ 2 



r 2i+ 1 

O /O i 



* 
1 



_ . 

Equating separately to zero the terms of the order , and those of the 
order a 2 , we shall have the two equations, 

, ^J* \ Y (*) = 2 U (1) __ -^(m 2 i) : 
2 i + i; 2 ^ 




C x being an arbitrary constant. The first of these equations detects Y 
and consequently the value of y. Substituting in the second member of 
the second equation, we shall develope by the method of No. 560. in a 
series of the form 

N<> + NW+ N^ + Sue. 

N (i) being subject to the same equation of partial differences as U w , and 
we shall determine the constant C in such a manner that N (0) is nothing; 
thus we shall have 

N 



Y & = 



4,* 



2i + 1 

and consequently 

The expression of the radius r of the surface of the fluid will thus be 
determined to quantities of the order a 3 , and we may, by the same process, 
carry the approximation as far as we wish. We shall not dwell any longer 
upon this object, which has no other difficulty than the length of calcula 
tions; but we shall derive from, the preceding analysis this important con 
clusion, namely, that we may affirm that the equilibrium is rigorously pos 
sible, although we cannot assign the rigorous figure which satisfies it ; for 
we may find a series of figures, which, being substituted in the equation of 
equilibrium, leave remainders successively smaller and smaller, and which 

become less than any given quantity. v 

T4 



29G A COMMENTARY ON [SECT. XII. & X1I1. 



COMPARISON OF THE PRECEDING THEORY WITH OBSERVATIONS. 

582. To compare with observations the theory we have above laid down, 
we must know the curve of the terrestrial meridians, and those which we 
trace by a series of geodesic operations. If through the axis of rotation 
of the earth, and through the zenith of a plane at its surface we imagine 
a plane to pass produced to the heavens; this plane will trace a great cir 
cle which will be the meridian of the plane : all points of the surface of 
the earth which have their zenith upon this circumference, will lie under 
the same celestial meridian, and they will form, upon this surface, a curve 
which will be the corresponding terrestrial meridian. 

To determine this curve, represent by u = the equation of the surface 
of the earth ; u being a function of three rectangular coordinates x, y, z. 
Let x , y , z , be the three coordinates of the vertical which passes through 
the place on the earth s surface determined by the coordinates x, y, z ; we 
shall have by the theory of curved surfaces, the two following equations, 

/d u\ , /du 

= 

0= 

Adding the first multiplied by the indeterminate >. to the second, we 
get 

dz - 



\dx 

This equation is that of any plane parallel to the said vertical : this ver 
tical produced to infinity coinciding with the celestial meridian, whilst its 
foot is only distant by a finite quantity from the plane of this meridian, 
may be deemed parallel to that plane. The differential equation of this 
plane may therefore be made to coincide with the preceding one by suita- 
blv determining the indeterminate X. 

I 

Let 

d z = a d x + b d /, 

be the equation of the plane of the celestial meridian ; comparing it with 
the preceding one, we shall get 



To get the constants a, b, we shall suppose known the coordinates of 



BOOK L] NEWTON S PRINCIPIA. 297 

the foot of the vertical parallel to the axes of rotation of the earth and that 
of a given place on its surface. Substituting successively these coordi 
nates in the preceding equation, we shall have two equations, by means of 
which we shall determine a and b. The preceding equation combined 
with that of the surface u = 0, will give the curve of the terrestrial meri 
dian which passes through the given plane. 

If the earth were any ellipsoid whatever, u would be a rational and 
entire function of the second degree in x, y, z ; the equation (a) would 
therefore then be that of a plane whose intersection with the surface of the 
earth, would form the terrestrial meridian : in the general case, this me 
ridian is a curve of double curvature. 

In this case the line determined by geodesic measures, is not that of 
the terrestrial meridian. To trace this line, we form a first horizontal 
triangle of which one of the angles has its summit at the origin of 
this curve, and whose two other summits are any visible objects. We de 
termine the direction of the first side of the curve, relatively to two sides 
of the triangle, and to its length from the point where it meets the side 
which joins the two objects. We then form a second horizontal triangle 
with these objects, and a third one still farther from the origin of the 
curve. This second triangle is not in the plane of the first; it has nothing 
in common with the former, but the side formed by the two first objects ; 
thus the first side of the curve being produced, lies above the plane of 
this second triangle; but we bend it down upon the plane so as always to 
form the same angles with the side common to the two triangles, and it is 
easy to see that for this purpose it must be bent along a vertical to this 
plane. Such is therefore the characteristic property of the curve traced 
by geodesic operations. Its first side, of which the direction may be 
supposed any whatever, touches the earth s surface; its second side is this 
tangent produced and bent vertically ; its third is the tangent of the se 
cond side bent vertically, and so on. 

If through the point where the two sides meet, we draw in the tangent 
plane at the surface of the spheroid, a line perpendicular to one of the 
sides, it is clear that it will be perpendicular to the other ; whence it follows? 
that the sum of the sides is the shortest line which can be drawn upon the 
surface between their extreme points. Thus the lines traced by geodesic 
operations, have the property of being the shortest we can draw upon the 
surface of the spheroid between any two of their points; andp.294,Vol.I. 
they would be described by a body moving uniformly in this surface. 



298 A COMMENTARY ON [SECT. XII. & XIII. 

Let x, y, z be the rectangular coordinates of any part whatever of the 
curve ; x + d x, y + d y, z + d z will be those of points infinitely near to 
it. Call d s the element of the curve, and suppose this element produced 
by a quantity equal tods; x + 2 d x, y + 2 d y, z + 2 d z will be the 
coordinates of extremity of the curve thus produced. By bending it ver 
tically, the coordinates of this extremity will become x + 2dx + d 2 x, 
y + 2 d y + d 2 y, z + 2dz-f<i 2 z; thus d 2 x, d 2 y, d - z 
will be the coordinates of the vertical, taken from its foot ; we shall there 
fore have by the nature of the vertical, and by supposing that u = is 
the equation of the earth s surface, 

/d u\ , /d 

= (die) 

d u 



/ u\ , 

--= (dx) d 



equations which are different from those of the terrestrial meridian. In these 
equations d s must be constant; for it is clear that the production of 
d s meets the foot of the vertical at an infinitely small quantity of the fourth 
order nearly. 

Let us see what light is thrown upon the subject of the figure of the earth 
by geodesic measures, whether made in the directions of the meridians, or in 
directions perpendicular to the meridians. We may always conceive an ellip 
soid touching the terrestrial surface at every point of it, and upon which, the 
geodesic measures of the longitudes and latitudes from the point of contact, 
for a small extent, would be the same as at the surface itself. If the entire 
surface were that of an ellipsoid, the tangent ellipsoid would every where 
be the same ; but if, as it is reasonable to suppose, the figure of the meri 
dians is not elliptic, then the tangent ellipsoid varies from one country to 
another, and can only be determined by geodesic measures, made in diffe 
rent directions. It would be very interesting to know the osculating ellip 
soids at a great number of places on the earth s surface. 

Let u = x " -\- y 2 + z 2 1 2 a u , be the equation to the surface 
of the spheroid, which we shall suppose very little different from a sphere 
whose radius is unity, so that a is a very small quantity whose square may 
be neglected. We may always consider u as a function of two variables 
x, y ; for by supposing it a function of x, y, z, we may eliminate z by 
means of the equation z = V I x 2 y 1 . Hence, the three equa 
tions found above, relatively to the shortest line upon the earth s surface, 
become 



BOOK I.] 



NEWTON S PRINCIPIA. 



299 






d 2 z zd 2 x = a - d J z ; 



yd 2 z zd a y = a (d~y) d z- 
This line we shall call the Geodesic line. 

Call r the radius drawn from the center of the earth to its surface, 6 the 
angle which this radius makes with the axis of rotation, which we shall 
suppose to be that of z, and p the angle which the plane formed by this 
axis and by r makes with the plane of x, y ; we shall have 

x = r sin. 6. cos. <f> ; y = r sin. 6 sin. p ; z = r cos. 6 ; 
whence we derive 

r 2 sin. 2 0. dp = xdy ydx; 

r 2 d 6 = (xdz zdx) cos. p + (y d z zdy) sin. p 
d s 2 = dx 2 +dy 2 +dz 2 = dr 2 +r 2 dd 2 +r 2 d p 2 sin. 6. 
Considering then u , as a function of x, y, and designating by -y the lati 
tude ; we may suppose in this function r= 1, and -y= 100 d, which gives 

x = cos. y cos. p ; y = cos. y- sin. p ; 
thus we shall have 

rd 



d u 



but we have 



-4/ 



= tan. <p ; 



cos. 



x 2 _J_ y 2 _ CQS 

whence we derive 

x d x + y d y , 

d 4/ = = r 1 - - r-; d 

sin. -y cos. -y 

Substituting these values of d -y and of d f in the preceding differential 
equation in u , and comparing separately the coefficients of d x and d y ; 
we shall have 

(d u \ _ cos. <p /d u\ sin. p 
d x / sin. 4/ \d -y / cos. 4/ 

/d u\ sin, g /d u\ 

\d y / ~ "* sin. 4 Vd^J H 
which give 



xdy ydx 

= - * ^ - 

x 2 



d u 
d~ 

cos. <p /d u 

cos. 



d u\ , 

-T )d 2 y 

d x/ ? 



dy 



^)d^ = - 
v / 



sin. -y cos. -y 
,du\ 


^** j y 

v -l_ ir fl 3 ^ \ 


^d p / | j t 



300 A COMMENTARY ON [SECT. XII. & XIII. 

But neglecting quantities of the order , we have x d 2 y y d 2 x = ; 
and the two equations 

xd 2 z zd 2 x = 0, yd 2 z zd 2 y = 0, 
give 

Z 2 (xd 2 x+yd 2 y) 

zd z = + / 

and 

x* + y 2 + z 2 = 1 
gives 

xd 2 x + yd 2 y + zd 2 z + tls 2 = 0; 
substituting for z d 2 z its preceding value, we shall have 

xd 2 x + yd 2 y = (x 2 + y 2 )ds 2 = d s 2 cos. 2 ^; 
wherefore 

d u \ , /d u \ , , /d u 



The first of equations (O), will thus give by integration, 

r 2 df sin. 8 * = cds+ ads/ds(|j-^ ); ..... (p) 

c being the arbitrary constant. 

The second of equations (O) gives 

d. (x d z z d x) =r a. (-, d " z ; 



-, 

but it is easy to see by what precedes, that we have 

d 2 z = d s 2 . sin. 4/ ; 
we have therefore 

d (x d z z d x) = ads ( - -\ sin. 4/ ; 
in like manner we have 

d (y d z z d y) = ads 2 f -,-- ^ sin. 4<; 

9 

we shall therefore have 

r " d = c d s sin. <p + c" d s cos. <p 

. C /d u\ /d u\ . , ) 

ads cos. <pf d s -( \.Td// OS * ^ "^" \d / Sm ^ ^ ( 

ads sin. p/d s| (~)sin. p (-j^)cos. p tan. -4/j; . (q) 

First consider the case in which the first side of the Geodesic line is 
parallel to the corresponding plane, of the celestial meridian. In this case 
d p is of the order , as also d r ; we rTave, therefore, neglecting quantities 
of the order a 2 , d s = r d 0, the arc s being supposed to increase from 



BOOK I.] NEWTON S PRINCIPIA. 301 

the equator to the poles. 4 1 expressing the latitude, it is easy to see that 
we have = 100 4/ (TTT) > which gives 

d0 = _d^-d 
we have therefore 



Thus naming s the difference in latitude of the two extreme points of 
the arc s, we shall have 



u/ being here the value of u at the origin of s. 

If the earth were a solid of revolution, the geodesic line would be al 
ways in the plane of the same meridian ; it departs from it if the parallels 
are not circles ; the observations of this deflection may therefore clear up 
this important point of the theory of the earth. Resume the equation (p) 
and observe that in the present case, d p and the constant c of this equa 
tion are of the order a, and that we may there suppose r = 1, d s = d -4/, 
6 = 100 -4/; we shall thus get 

d <p cos. 2 -\}/ = cd-vJ/ + a 



However, if we call V the angle which the plane of the celestial meri 
dian makes with that of x, y, whence we compute the origin of the angle 
<p; we shall have d x = tan. V = d y ; x , y , z being the coordinates 
of that meridian whose differential equation, as we have seen in the pre 
ceding No., is 

d z = a d x + b d y . 

Comparing it with the preceding one, we see that a, b are infinite and 

Q 

such that -- p = tan. V, the equation (a) of the preceding No. thus 

gives 

/d u\ , /d u 

0=(^-).tan.V-(^ 

whence we derive 



We may suppose V = <p, in the terms multiplied by u; moreover 



= tan. <p : w have therefore 
x 



302 A COMMENTARY ON [SECT. XII. & XIII. 

/du\ 

\T / 
cos. 4> cos. <f> tan. <p tan. V] = 



cos. 4* cos. <f 
which gives 



. 

cos. 2 4 

The first side of the Geodesic line, being supposed parallel to the plane 
of the celestial meridian, the differentials of the angle V, and of the dis 
tance (<f> V) cos. 4- from the origin of the curve to the plane of the 
celestial meridian ought to be nothing at this origin ; we have therefore 
at this point 

a( j ) 

\d / 



tan. 



~^, 
cos. 2 4 

and consequently, the equation (p) gives 



u, and 4 / being referred to the origin of the arc s. 

At the extremity of the measured arc, the side of the curve makes with 
the plane of the corresponding celestial meridian an angle very nearly 
equal to the differential of (p V) cos. 4 / > divided by d 4^ V being sup 
posed constant in the differentiation ; by denoting therefore this angle by 
, we shall have 

d / TT\ 

a ~ cos. 4 (<p V) sin. 4 / - 

If we substitute for -~- its value obtained from the equation (p), and for 

f V, its preceding value, we shall have 

a f /d u/\ . /d u \ , , /d u \ ") 

= -- . < ( T-*-I tan. 4^/ [~i ) tan. 4 / + / d 4^ ( -i ) / ; 
cos. <p \\ d <p J Wlp/ r V.d?>/J 

the integral being taken from the origin of the measured arc, to its extre 
mity. Call s the difference in latitude of its two extreme points ; being 
supposed sufficiently small for t z to be rejected, we shall have 

a E tan. 4 / /d u\ / d 2 u \ \ 

w = -- -Z -! ( > tan. 4 + ( T - 1 r ) ( 5 
cos. 4/ I \d p / r \d p d 4// J 

in which the values of 4> TT ^? and f -, - y r^must be referred, for the 
*\ dp/ \d f d 4// 

greater exactness, to the middle of the measured arc. The angle -a must be 



BOOK I.] NEWTON S PRINCIPIA. 303 

supposed positive, when it quits the meridian, in the direction of the in 
crements of <p. 

To obtain the difference in longitude of the two meridians correspond 
ing to the extremities of the arc, we shall observe, that u/, V,, -4/,, and 
p /5 being the values of u , V, 4/, and <p, at the first extremity, we have 



, v .= 



d u/x /d u 

d p 



/d u\ 

vd/ 



cos. 2 ^ cos. 2 -4, 

but we have very nearly, neglecting the square of , 

c s /d u/x 

c = a -r- tan. ; 



cos/ s 

1 * 

we shall have, therefore, 

V-V = ^-p. ((^tan. 

cos. Y (_ d p / 

whence results this very simple equation, 

(V V,) sin. -4>, = ; 

thus we may, by observation alone, and independently of the knowledge 
of the figure, determine the difference in longitude of the meridians cor 
responding to the extremities of the measured arc ; and if the value of the 
angle -a is such that we cannot attribute it to errors of observations, we 

O 

shall be certain that the earth is not a spheroid of revolution. 

Let us now consider the case where the first side of the Geodesic line 
is perpendicular to the corresponding plane of the celestial meridian. If 
we take this plane for that of x, y, the cosine of the angle formed by this 

side upon the plane, will be C X . 2 + ; thus this cosine being no 
thing at the origin, we have d x = 0, d z = 0, which gives 

d . r sin. 6 cos. <f> = ; d . r cos. 6 = ; 
and consequently 

r d 6 = r d <p sin. 6 . cos. 6 . tan. <p ; 

but we have, to quantities of the order a % d s = r d o sin. 6 ; we shall 
have, therefore, at the origin, 

d d _ tan. <p . cos. 6 
d s r 

The constant c", of the equation (q), is equal to the value of x d z 
z d x, at the origin ; it is therefore nothing, and the equation (q) gives at 
the origin, 

-i = r sin. <p ; 
d s r 2 



304 A COMMENTARY ON [SECT. XII. & XIII. 

we have, therefore, observing that p is here of the order a, and that thus 
neglecting quantities of the order a 2 , we have sin. <p = tan. p, 

c = r, cos. O fl 

the quantities r, and 6 / being relative to the origin ; therefore, if we con 
sider that at this origin the angle p is what we have before called it, 

/d U/v 

\"Ty 

p, V,, and whose value we have found equal to 2 ; we shall 

have at this point 



The equation (q) then gives 

d l 
d s 2 

but we have 



dd, /d u \ sin. -J/. 

36 ss a I i - 

d s \ d p / cos. 2 4v 

ives 

. cos. 0. d p. /d u/\ 

/ f ff I / I 

2 - r y ds " \~d^J 



--j / 
d s r x sin. ^ 

we shall get therefore 

^- = (1 - 2 . u/) tan. +/ + . (^) tan. +, 
Observing that at the origin, 

au 



-~ - -. - 

d s r 7 sin. ^ y cos. - 

the equation (p) gives 

c = r, sin. 6, ; 
whence we get 

d u/ . d 0. /d u/\ 

j . 2 a . -=-*- 2 . T cos. 6, a ( -=-i- ) 

d 2 p x d s d s _ \ d p/ 

d s 2 " r 7 sin. 6, r, sin. e 0, cos. 2 ^ 

and consequently 

d 8 p, /d u/\ 2 cos. * ^ 

d s 2 v d p / cos. 4 4-, 

The equ|tion 



gives, by retaining amongst the terms of the order s 2 , only those which are 
independent of a, 

II 14 i 2 dJJ/ as / d 2 u/ x 

"* " S ds "" 2 S ds 2 " cos. 4/VdpdV 



BOOK I.] NEWTON S PRINCIPIA. 305 

wherefore 

1 i s f /d u/\ / d 2 u 

- 



The difference ofktitudes at the two extremities of the measured arc, 
will therefore give 



It is remarkable, that for the same arc, measured in the direction of the 

meridian, this function, by what precedes, is equal to - ; it may thus 

tan. y / * 

be determined in two .ways, and we shall be able to judge whether the 
values thus found of the difference of latitudes, or of the azimuthal 
angle *-, are due to the errors of observations, or to the excentricity of the 
terrestrial parallels. 

Retaining only the first power of s, we have 



9 <?< is not the difference in longitude of the two extremities of the arc 
s ; this difference is equal to V V, ; but we have, by what precedes, 



cos. 
which gives 



V (d> -- V) - 



os., cos. 

wherefore 



For greater exactness, we must add to this value of V _ V 7 the term 
depending on s 3 , and independent of a, which we obtain in the hypothesis 
of the earth being a sphere. This term is^ equal to A s 3 . 
thus we have 



tan> 
cos. 



It remains to determine the azimuthal angle at the extremity of the 
arc s. For that purpose, call x , and y , the coordinates x, y, referred to 



VOL. 1J. 



306 A COMMENTARY ON [SECT. XII. & XIII. 

the meridian of the last extremity of the arc s ; it is easy to see that the 

V d x 2 + d z 2 

cosine of the azimuthal anjjle is equal to \ . If we refer 

d s 

the coordinates x, y, to the plane of the meridian corresponding to the 
first extremity of the arc ; its first side being supposed perpendicular to 
the plane of this meridian, we shall have 

*2i o-i^ n.l 
d s d s - d s 

wherefore, retaining only the first power of s, 

d_x d 2 x, dj5 d^_z, 

d7 : * Ts^ dl - s dT e5 

but w r e have 

x = x cos. (V V,) + y sin. (V V,) ; 

thus V V, being, by what precedes, of the order a, we shall have 

"dT = S ~d7^ + ( V ~~ V )~dT* 
Again, we have 

x = r sin. 6 cos. p ; z = r cos. & ; 

we therefore shall obtain, rejecting quantities of the order a 2 , and observ 
ing that p., -T /, and j are quantities of the order , 
d s d s 

d 2 x. d 2 u/ . d 2 0. d <?* 

, f = a . , % sin. 6 + r . -j { cos. 6 r sin. ^ . - .- - . 
ds 2 ds 2 ds 2 ds 8 

Thence we have 



d u/ /d 2 u/\dp, 8 
- ~ ^ 



)up / /uu/x u / 

d s z Wl-vp/ds 2 cos. 
moreover, d s = r, sin. 6 / . d tp , ; we shall, therefore, have by substitutijig 

dp. , d 2 d, , . ,. 

for r,, 0,, -r 1 , and -, -, their preceding values, 
d s d s 2 

d 2 x ., sin. ~ -^f>. /d u/\ 

i zz ( 1 a u/) r~ + a ( j r ) tan. z -J/, sin. Jc. 

d s z cos. ^ \d -y / 

r 1 1 a u. + a \-\ -f- ) tan. -vL. f + 

cos. v/ *- Ml Y / cos. 

Neglecting the superior powers of s, we have, as we have seen, 
V V ? J /d u 

T ~ . "t\l /. /*-* U /ti 1 

COS. Y/ f a l1 / "T a \T r" J ^ an - "r. 

and - - = 1 ; we therefore have 



BOOK I.] NEWTON S PRINCIPIA. 307 

x / /i A 8 - 2 ^/, /"du/v . /d 2 u/x sin. * 4. 

- = s(l au/) r^-f-as( , / )tan. 2 4..sin. 4, s(-, M r-f/j 

s x cos. 4y vdJ*/ v d p 2 / cos. 3 4, 



dx, ,. A sin. 2 4 /du 

-, 

ds 

in like manner we shall find 
dz . . 



the cosine of the azimuthal angle, at the extremity of the arc s, will thus 
be 

s tan. 

This cosine being very small, it may be taken for the complement of 
the azimuthal angle, which consequently is equal to 

100 s tan 



( r d u Ai 

. 4X , , , /du/x . \dp*} V. 

V - a <+H^) tan ^--^sTM7/ 



C /d 2 u/x -\ 

w , , , /du/N f , a Vdp 2 ; v. 

T/ J 1 a u/+a ( -j-^- ) tan. ^ i ! ^ 1 ( 

I. \d 4 / cos. 2 4-, ) 



For the greater exactness, we must add to this angle that part depend 
ing on s 3 , and independent of , which we obtain in the hypothesis of the 
earth s sphericity. This part is equal to s 3 (| + tan. z 4 1 /) tan. 4 1 /, Thus 
the azimuthal angle at the extremity of the arc s is equal to 



100-stan.4 



The radius of curvature of the Geodesic line, forming any angle what 
ever with the plane of the meridian, is equal to 

ds 2 

V (d 2 x) 2 + (d 2 y) 2 + (d*z) 2 

d s being supposed constant; let R be this radius. The equation 
x 2 + y2 + z 2 = l + 2u / gives 

xd 2 x+ yd 2 y + zd 2 z = d s 2 + ad 8 u ; 

if we add the square of this equation to the squares of equations (O), we 
shall have, rejecting terms of the order a 2 , 

(x+ y 2 + z 2 ) (d 2 x) 2 + (d 2 y) 2 + (d 2 z) 2 }=ds 4 2ads 2 d 2 u 

whence we derive 

d 2 u 
R = 1 + au + a-^y. 

In the direction of the meridian, we have 

d ! u 



wherefore 



U2 



308 A COMMENTARY ON [SECT. XII. & XIII. 

In the direction perpendicular to the meridian, we have by what pre 
cedes, 



wherefore 

rd 2 u 



R = 



d , 

c 

If in the preceding expression of V V, , we make -^5- = s , it takes 
this very simple form relative to a sphere of the radius R, 

V V, = ^r-. -f 1 l-s fz . tan. 8 >}>,{. 

cos. 4V i. J 

The expression of the azimuthal angle becomes 

100 s tan. -4>, [I i s /2 (J + tan. 2 ^ / }}. 

Call X, the angle which the first side of the Geodesic line forms with the 
plane corresponding to the celestial meridian, we shall have 



u /u p _ 

2 = Vd^J dT 2+ Vdf^/ dV + Wl p 2 ^ ds 2-t \dpdV ds d s^ Vd^V d s 2 

But supposing the earth a sphere, we have 

dft _ sin. X . d. p, _ 2 sin. X cos. X ^ ^ . 
d s ~ cos. ^ d s 2 " cos ^ 



wherefore, 



5 - = cos. X ; = sin. 2 X tan. 

d s d s 2 



_ sn... tan 

" 



_ 

ds 2 " cos. 



the radius of curvature R, in the direction of this Geodesic line, is there 
fore 



To abridge this, let 

d 2 U 



K = 



r 



BOOK I.] NEWTON S PRINCIPIA. 309 

A = 




R = K + A sin. 2 X + B cos. 2 X. 

The observations of azimuthal angles, and of the difference of the lati 
tudes at the extremities of the two geodesic lines, one measured in the 
direction of the meridian, and the other in the direction perpendicular to 
the meridian, will give, by what precedes, the values of A, B and K ; for 
the observations give the radii of curvature in these two directions. Let 
R, and R be these radii ; we shall have 

R + R" 

~2~ 
R R R" 

B= ~~2 - ; 

and the value of A will be determined, either by the azimuth of the ex 
tremity of the arc measured in the direction of the meridian, or by the 
difference in latitude of the two extremities of the arc measured in a di 
rection perpendicular to the meridian. We shall thus get the radius of 
curvature of the geodesic line, whose first side forms any angle whatever 
with the meridian. 

j 
If we call 2 E, an angle whose tangent is-^-, we shall have 



R = K + VA" + B 2 . cos. (2 X - 2 E) ; 

the greatest radius of curvature corresponds with X =r E ; the correspond 
ing geodesic line forms therefore the angle E, with the plane of the me 
ridian. The least radius of curvature corresponds with X = 100+ E; 
let r be the least radius, and r the greatest, we shall have 

R = r + (r r) cos. 2 (X E), 

X E being the angle which the geodesic line corresponding to R, forms 
with that which corresponds with r . 

We have already observed, that at each point of the earth s surface, 
we may conceive an osculatory ellipsoid upon which the degrees, in all 
directions, are sensibly the same to a small extent around the point of os 
culation. Express the radius of this ellipsoid by the function 

1 a sin. 2 %J/ Jl + h cos. 2 (<p + /3)j, 

the longitudes <p being reckoned from a given meridian. The expression 

us 



310 A COMMENTARY ON [SECT. XII. & XIII. 

of the terrestrial meridian measured in the direction of the meridian, 
will be, by what precedes, 

^ . [I + h cos. 2 (<p + /5)J . { 1 + 3 cos. 2 -4/ 3 * sin. 2 4}. 

If the measured arc is considerable, and if we have observed, as in 
France, the latitudes of some points intermediate between the extremity; 
we shall have by these measures, both the length of the radius taken for 
unity, and the value of {1 + h cos. 2 (p + 13)}. We then have, by 
what precedes, 

, tan. 8 4 (1 + cos. 2 40 . . 

= 2 h . E . i *. . sin. 2 (a + ) ; 

cos. 4 

the observation of the azimuthal angles at the two extremities of the arc 
will give a h sin. 2 (<p + /3). Finally, the degree measured in the direc 
tion perpendicular to the meridian, is 

1 + 1. ajl + h cos. 2 (<f> + 13)} sin. 2 4 + 4. ah tan. 2 $ cos. 2 (<p + ); 
the measure of this degree will therefore give the value of h sin. 2 (p + ,6). 
Thus the osculatory ellipsoid will be determined by these several mea 
sures : it would be necessary for an arc so great, to retain the square of e 
in the expression of the angle ; and the more so, if, as it has been ob 
served in France, the azimuthal angle does not vary proportionally to 
the measured arc: at the same time we must add a term of the form 
k sin. 4- cos. 4/ sin. (<p -f /3 ), to get the most general expression of this 
radius. 

583. The elliptic figure is the most simple after that of the sphere : we 
have seen above that this ought to be the figure of the earth and planets, 
on the supposition of their being originally fluid, if besides they have 
retained their primitive figure. It was natural therefore to compare 
with this figure the measured degrees of the meridian; but this compari 
son has given for the figure of the meridians different ellipses, and which 
disagree too much with observations to be admissible. However, before we 
renounce entirely the elliptic, we must determine that in which the greatest 
defect of the measured degrees, is smaller than in every other elliptic 
figure, and see whether it be within the limits of the errors of observations. 

O 7 

We arrive at this by the following method. 

Let a (1) , a (2) , a (3) , &c. be the measured degrees of the meridians ; p (1) , 
p (2 \ p C3) , &c. the squares of the sines of the corresponding latitudes : 
suppose that in the ellipse required, the degree of the meridian is expressed 
by the formula z + p y ; calling x (1) , x (2) , x (3 >, &c. the errors of observation, 
we shall have the following equations, in which we shall suppose that p (1) s 
p v % p , &c. form an increasing pi-ogression, 



BOOK I.] NEWTON S PRINCIPIA. 311 

a") z pOy = x) 

a ( 2 ) z p>y = x< 2 > ...... (A) 



a (n) z p(n) y __ x (n) 

n being the number of measured degrees. 

We shall eliminate from these equations the unknown quantities z and y, 
and we shall have n 2 equations of condition, between the n errors 

x (1) , x (2 -, x (n) . We must, however, determine that system of errors, 

in which the greatest, abstraction being made of the signs, is less than in 
every other system. 

First suppose that we have only one equation of condition, which may 
be represented by 

a = m x (1 ) + n x {2 > + p x f3) + &c. 

a being positive. We shall have the system of the values of x (1) , x (2) , &c. 
which gives, not regarding signs, the least value to the greatest of them ; 
supposing them all nearly equal, and to the quotient of a divided by the 
sum of the coefficients, m, n, p, &c. taken positively. As to the sign 
which each quantity ought to have, it must be the same as that of its co 
efficient in the proposed equation. 

If we have two equations of condition between the errors, the system 
which will give the smallest value possible to the greatest of them will be 
such that, signs being abstracted, all the errors will be equal to one ano 
ther, with the exception of one only which will be smaller than the rest, 
or at least not greater. Supposing therefore that x (1) is this error, we 
shall determine it in function x (2) , x (3) , &c. by means of one of the proposed 
equations of condition ; then substituting this value of x (1) in the other 
equation of condition, we shall form one between x {2) , x (3) , &c. ; which re 
present by the following 

a = m x + n x (3 > + &c. 

a being positive; we shall have, as above, the values of x (2) , x (3) , &c. by 
dividing a by the sum of the coefficients m, n, &c. taken positively, and by 
giving successively to the quotient the signs ofm, n, &c. These values sub 
stituted in the expression of x (1) in terms of x ( % x (3) , &c. will give the value 
of x U) ; and if this value, abstracting signs, is not greater than that of x (2) , 
this system of values will be that which we must adopt; but if greater, then 
the supposition that x (1) is the least error, is not legitimate, and we must 
successively make the same supposition as to x (2 >, x (3) , &c. until we arrive 
at an error which is in this respect satisfactory. 

If we have three equations of condition between the errors ; the system 
which will give the smallest value possible to the greatest of them, will be 

U4 



312 A COMMENTARY ON [SECT. XII. & XIII. 

such, that, abstracting signs, all the errors will be equal, with exception of 
two, which will be less than the others. 

Supposing therefore that x (1) , x ( 2) are these two errors, we shall elimi 
nate them from the third of the equations of condition by means of the 
other two, and we shall have an equation between the errors x (3) , x (1) , &c.: 
represent it by 

a = m x + n x W> + &c. 

a being positive. We shall have the values of x (3) , x (1) , &c. by dividing 
a by the sum of the coefficients m, n, &c. taken positively, and by giving 
successively to the quotient, the signs of m, n, &c. These values substi 
tuted in the expressions of x (1) , and of x (2) in terms of x , x W, &c. will 
give the values of x (1) , and of x (2) , and if these last values, abstracting 
signs, do not surpass x (3 \ we shall have the system of errors, which we 
ought to adopt; but if one of these values exceed x (3) , the supposition that 
x (1) , and x ^ are the smallest errors is not legitimate, and we must use 
the same supposition upon another combination of errors x (1) , x (2) , c. 
taken two and two, until we arrive at a combination in which this suppo 
sition is legitimate. It is easy to extend this method to the case where 
we should have four or more equations of condition, between the errors x (l \ 
x (2) , &c. These errors being thus known, it will be easy to obtain the 
values of z and y. 

The method just exposed, applies to all questions of the same nature ; 
thus, having the number n of observations upon a comet, we may by this 
means determine that parabolic orbit, in which the greatest error is, ab 
stracting signs, less than in any other parabolic orbit, and thence recog 
nise whether the parabolic hypothesis can represent these observations. 
But when the number of observations is considerable, this method be 
comes too tedious, and we may in the present problem, easily arrive at 
the required system of errors, by the following method. 

Conceive that x (i) , abstracting signs, is the greatest of the errors 
x (1) , x , &c. ; we shall first observe, that therein must exist another error 
x (l \ equal, and having a contrary sign to x (i) ; otherwise we might, by 
making z to vary properly in the equation 

a (i) z p (i) . y = x w, 

diminish the error x w , retaining to it the property of being the extreme 
error, which is against the hypothesis. Next we shall observe that x w 
and x (i/) being the two extreme errors, one positive, and the others nega 
tive, and equal to one another, there ought to exist a third error x (l ">, 
equal, abstracting signs, to x (i) . In fact, if we take the equation corre- 



BOOK I.] NEWTON S PRINCIPIA. 313 

spending to x (i , from the equation corresponding to x (i/) , we shall 
have 

a 00 _ a W {p V _ p WJ. y - x M x . 

The second member of this equation is, abstracting signs, the sum of 
the extreme errors, and it is clear, that in varying y suitably, we may di 
minish it, preserving to it the property of being the greatest of the sums 
which we can obtain by adding or subtracting the errors x (1) , x (2) , &c. 
taken two and two ; provided there is no third error x (i "> equal, abstract 
ing signs, to x (i > ; but the sum of the extreme errors being diminished, 
and these errors being made equal, by means of the value of z, each of 
these errors will be diminished, which is contrary to the hypothesis. 
There exists therefore three errors x (i) , x (i/) , x (i//) equal to one another, 
abstracting signs, arid of different signs the one from the other two. 

Suppose that this one is x ^ ; then the number i will fall between the 
two numbers i and i". To show this, let us imagine that it is not the 
case, and that i is below or above both the numbers i, i". Taking the 
equation corresponding to V, successively from the two equations corre-^ 
spending to i and to i", we shall have 

a 0) _ a M (p _ p(i )) y - x W _ X M; 
a G") _ a GO (p (i "> p (i/) ) y = x ^ x . 

The second members are equal and have the same sign ; these are also, 
abstracting signs, the sum of the extreme errors; but it is evident, that 
varying y suitably, we may diminish each of these sums, since the coeffi 
cient of y, has the same sign in the two first members : moreover, we may, 
by varying z properly, preserve to x (i/) the same value; x w and x (i ;) will 
therefore then be, abstracting signs, less than x (i/) which will become the 
greatest of the errors without having an equal ; and in this case, we may, 
as we have seen, diminish the extreme error ; which is contrary to the hy 
pothesis. Thus the number i 7 ought to fall between i and i". 

Let us now determine which of the errors x (1) , x (2) , &c. are the extreme 
errors. For that purpose, take the first of the equations (A) successively 
from the following ones, and we shall have this series of equations, 
a w _ a w (p< 2 ) p (1) ) y = x (2 > x<, 
a C3)__ a (1 > (p (3 > p>) y = x 13 x; . . . . (13) 
&c. 

Suppose y infinite ; the first members of these equations will be nega 
tive, and then the value of x (n will be greater than x (2) , x (3) , &c. : dimin 
ishing y continually, we shall at length arrive at a value that will render 
positive one of the first members, which, before arriving at this state, will 



314 A COMMENTARY ON [SECT. XII. & XIII. 

be nothing. To know which of these members first becomes equal to zero, 
we shall form the quantities, 

a (2)_ a U) a (3)_ a (D a (4)_ a (D 



p u; " p w _ p u; p w P 

o (r) _ o 



_p(D 

Call ]3 (1 ^ the greatest of these quantities, and suppose it to be 



if there are many values equal to /3 W , we shall consider that which cor 
responds to the number r the greatest, substituting (3 (l i for y, in the 
(r l) th of the equations (B), x (r) will be equal to x (1 \ and diminishing 
y, it will be equal to x (1) , the first member of this equation then becoming 
positive. By the successive diminutions of y, this member will increase 
more rapidly than the first members of the equations which precede it ; 
thus, since it becomes nothing when the preceding ones are still nega 
tive, it is clear that, in the successive diminutions of y, it will always be 
the greatest which proves that x (r > will be constantly greater than x (1) , 
x (2) , . . . x^- 1 ), when y is less than /S (1 ). 

The first members of the equations (B) which follow the (r l) th will 
be at first negative, and whilst that is the case, x (r + 1) , x (r + 2) , &c. will be 
less than x (1) , and consequently less than x (r) , which becomes the greatest 
of all the errors x (I) , x (2) , .. . x ( "), when y begins to be less than /3 (1) . But 
continuing to diminish y, we shall get a value of it, such that some of the 
errors x (r + J) , x (r + 2) , &c. begin to exceed x (r) . 

To determine this value of y, we shall take the r th of equations (A) suc 
cessively from the following ones, and we shall have 

a (r + D _ a (r) _ Jp(r + l) _ p (r)J y _ x (r + 1) __ x (r) . 
a (r + 2) a (r) Jp(r + 2) p WJ y x (r + 2) x (r)^ 

Then we shall form the quantities 

a (r + 1) a (r) a (r + 2) a (r) 

5irp 

p(r+l) pW p(r + 2) p(r) 

Call /3, the greatest of these quantities, and suppose that it is 
." (r/ ~ . : if many of these quantities are equal to (3 (2) , we shall suppose 

that r is the greatest of the numbers to which they correspond. Then x W 
will be the greatest of the errors x (1) , x (2) , &c. . . . x (n) so long as y is com 
prised between /3 (1) , and /3 (2) ; but when by diminishing y, we shall arrive at 
6 (2) ; then x (r/) will begin to exceed x w , and to become the greatest of the 
errors. 

To determine within what limits we shall form the quantities 



Let /3 W > be the greatest of these quantities, and suppose that it is 



BOOK I.] NEWTON S PRINCIPIA. 315 



^ _ 

(1 . j* - <yj : if several of the quantities are equal to /3 ( 3 \ we shall sup 

pose that r" is the greatest of the numbers to which they correspond, x (r ) 
will be the greatest of all the errors from y = /3^, to y = /3( 3 >. When 
y = /3( 3 ), then x (r ") begins to be this greatest error. Thus preceding, we 
shall form the two series, 



oo; j8W; J3; /SC 3 ) ; . . . /3D ;_a> ; ..... (C) 
The first indicates the errors x (1) , x (r \ x 1 ^, &c. which become succes 
sively the greatest : the second series formed of decreasing quantities, in 
dicates the limits of y, between which these errors are the greatest; thus, 
x^ is the greatest error from y = cc, to y = j8W ; x W is the greatest er 
ror from y = (3( l \ to y = /3 (2) ; x^ is the greatest error from y = /3^, 
to y = /3 (3) , and so on. 

Resume now the equations (B) and suppose y negative and infinite. 
The first members of these equations will be positive, x ^ will therefore then 
be the least of the errors x^, x (2) , &c. : augmenting y continually, some 
of these members will become negative, and then x (1) will cease to be the 
least of the errors. If we apply here the reasoning just used in the case 
of the greatest errors, we shall see that if we call xW the least of the 
quantities 



a (s) a (l) 

and if we suppose that it is } ^ , s being the greatest of the num 
bers to which X( l ) corresponds, if several of these quantities are equal to 
XOj x (i) -will b e the least of the errors from y = oc, to y = X( \ In 
like manner if we call X( 2 the least of the quantities 

a( s + ! ) a^ a (s + 2) a (s) 
( , + (5)5 (8 + 2) ft , &c. 

O (** / _^ *1 (*V 

and suppose it to be T^ ^j , s being the greatest of the numbers to 

which X< 2 ) corresponds, if several of these quantities are equal to xW; x^ 
will be the smallest of the errors from y = X^, to y r= X^; and so forth. 
In this manner we shall form the two series 

x jx ;x ;x - ;...x^ p 

x; XO; xW; XW;...X^; oo ; (D) 

The first indicates the errors xW, x<">, x^ 8 >, &c. which are successively 
the least as we augment y : the second series formed of increasing terms, 
indicates the limits of the values of y between which each of these errors 



316 A COMMENTARY ON [SECT. XII. & XIII. 

is the least; thus x ri > is the least of the errors fromy = GO, to y = X^ 
x (s) is the least of the errors, from y = X^, to y = X ( % and thus of the 
rest. 

Hence the value of y which, to the required ellipse, will be one of the 
quantities /3W, jSW, j3^; &c. X^, X< 2 >, &c. ; it will be in the first series, 
if the two extreme errors of the same sign are positive. In fact, these 
two errors being then the greatest, they are in the series x^, x w , x w , 
&c. ; and since one and the same value of y renders them equal they 
ought to be consecutive, and the value of y which suits them, can only 
be one of the quantities /3^, /S^, &c. ; because two of these errors cannot 
at the same tune be made equal and the greatest, except by one only of 
these quantities. Here, however, is a method of determining that of the 
quantities $^\ /3( 2) , &c. which ought to be taken for y. 

Conceive, for example, that /3 (3) is this value; then there ought to be 
found by what precedes between x^, and x (l % an error which will be the 
minimum of all the errors, since x (r/) , and x (l " ; will be the maxima of these 
errors; thus in the series x^, x^, x^ -*, &c. soma one of the numbers 
s, s , &c. will be comprised between r and r . Suppose it to be s. That 
x (s) may be the last of the value of y, it ought to be comprised between 
X (1) and X (2 > ; therefore if |3 is comprised by these limits, it will be the 
value sought of y, and it will be useless to seek others. In fact, suppose 
we take that of the equations (A), which answers to x (s) successively from 
the two equations which respond to x tr/) and to x (r " } ; we shall have 
ado _ a W {p p M} y = x< r > x ; 

a (r") _ a (s) _ lp (r") _ p (s)^y = - x (r") _ x (s). 

All the members of these equations being positive, by supposing 
y = ft (3) , it is clear, that if we augment y, the quantity x (r/) x (s) will 
increase ; the sum of the extreme errors, taken positively, will be there 
fore augmented. If we diminish y, the quantity x (r ") x (s) will be aug 
mented, and consequently also the sum of their extremes ; /3 (3 > is therefore 
the value of y, which gives the least of these sums; whence it follows that 
it is the only one which satisfies the problem. 

We shall try in this way the values of /3 (1) , j3 (2) , (3 W, &c., which is easily 
done by inspection ; and if we arrive at a value which fulfils the preced 
ing conditions, we shall be assured of the value required of y. 

If any of these values of j8 does not fulfil these conditions, then this 
value of y will be some one of the terms of the series x^, X C2) , &c. Con 
ceive, for example, that it is X , the two extreme errors x Cs ) and x ^ will 
then be negative, and it will have, by what precedes, an intermediate error, 



BOOK I.] NEWTON S PRINCIPIA. 317 

which will be a maximum, and which will fall consequently in the series 
x (1) , x (r \ x (r/ ), & c . Suppose that this is x to, r being then necessarily 
comprised between s and s j X (2 ) ought, therefore, to be comprised be 
tween j8 (I > and /S (2) . If that is the case, this will be a proof that X is the 
value required of y. We shall try thus all the terms of the series X< 2 ), x, 
X W, &c. up to that which fulfils the preceding conditions. 

When we shall have thus determined the value of y, we shall easily ob 
tain that of z. For this, suppose that J3 (2 > is the value of y, and that the 
three extreme errors are x (r) , x (r/) , x (s) ; we shall have x (s) = x to ? and 
consequently 

a (r) z p (r). y X to 

a to z p (s) . y = x (r) ; 
whence we get 

to + a (s) p (r) + p (s) 



rj _ __ 

Z _ 



2 2 

then we shall have the greatest error x W 5 by means of the equation 
a W _ a to p (s) _ p (r) 

X to = _1_ . E. v. 

2 2 y 

584. The ellipse determined in the preceding No. serves to recognise 
whether the hypothesis of an elliptic figure is in the limits of the errors of 
observations ; but it is not that which the measured degrees indicate witli 
the greatest probability. This last ellipse, it seems, should fulfil the 
following conditions, viz. 1st, that the sum of the errors committed in the 
measures of the entire measured arcs be nothing : 2dly, that the sum of 
these errors, all taken positively, may be a minimum. Thus considering 
the entire ones instead of the degrees which have thence been deduced, 
we give to each of the degrees by so much the more influence upon the 
ellipticity which thence results for the earth, as the corresponding arc is 
considerable, as it ought to be. The following is a very simple method 
of determining the ellipse which satisfies these two conditions. 

Resume the equations (A) of 589, and multiply them respectively 
by the numbers which express how many degrees the measured arcs 
contain, and which we will denote by i (l \ i (2) , i (3) , &c. Let A be the sum 
of the quantities i (1) . a (1) , i (2) . a (2) , &c. divided by the sum of the numbers 
i (1) , i (2) , &c. ; let, in like manner, P denote the sum of the quantities 
i (1) . p (1 >, i (2 l p (2 ), &c. divided by the sum of the numbers i W, i (2) , &c. ; 
the condition that the sum of the errors i (1) . x (l) , i (2) . x ( -\ &c. is nothing, 
gives 

= A z P.y. 




318 A COMMENTARY ON [SECT. XII. XIII. 

If we take this equation from each of the equations A of the preceding 
No., we shall have equations of the following form : i 






b (1 > b 
Form the series of quotients ^ , ^ , &c. and dispose them according 

to their order of magnitude, beginning with the greatest ; then multiply 
the equations O, to which they respond, by the corresponding numbers 
i (1) , i (2) , &c. ; finally, dispose these thus multiplied in the same order as 
the quotients. 

The first members of the equations disposed in this way, will form a 
series of terms of the form 

hWy c; h^y c; hy c^;&c. . . . (P) 
in which we shall suppose h (1) , h ^ positive, by changing the sign of the 
terms when y has a negative coefficient. These terms are the errors of 
the measured arcs, taken positively or negatively. 

Then it is evident, that in making y infinite, each term of this series 
becomes infinite ; but they decrease as we diminish y, and end by being 
negative at first, the first, then the second, and so on. Diminishing y 
continually, the terms once become negative continue to be so, and de 
crease without ceasing. To get the value y, which renders the sum of 
these terms all taken positively a minimum, we shall add the quantities 
h (1) , h (2) , &c. as far as when their sum begins to surpass the semi-sum of 
all these quantities ; thus calling F this sum, we shall determine r such 

that 

+ h< 2 > + h^ + ____ + h > 3 F; 

---- + h^- ) < F. 



C (r) 

We shall then have y = r-^-, , so that the error will be nothing rela 

tively to the same degree which corresponds to that of the equations (O), 
of which the first member equated to zero, gives this value of y. 

To show this, suppose that we augment y by the quantity 3 y, so that 

c W c (r - ) c fr ) 

r } + 3 y may be comprised between (r | j and j - } . The (r 1) first 

c 0> 
terms of the series (P) will be negative, as in the case of y = rrrj; hut in 

taking them with the sign +, their sum will decrease by the quantity 
jhd) + hOO ____ h ( - )} 3y. 



BOOK I.] NEWTON S PRINCIPIA. 319 

c (>-) 
The first term of this series, which is nothing when y = T-T- , will be 

come positive and equal to h ^ d y ; the sum of this term and the follow 
ing, which are positive, will increase by the quantity 

{hW+ h< r + + &c.} ay; 
but by supposition we have 

hO) + hw ____ h r - > < h + h ( + " + &c. ; 

the entire sum of the terms of the series (P), all taken positively, will 
therefore be augmented, and as it is equal to the sum of the errors 
i(i). x (| ) + i&. x (2) , &c. of the entire measured arcs, all taken with the 

c () 

sign + , this last sum will be augmented by the supposition of y = r-^ } + & y. 

It is easy to prove, in the same way, that by augmenting y, so as to be 

c (r-l) c (r-2j c (r-2) c (r - 3) 

comprised between , - n and T- - -. , or between 7-7- ^ and ,-7- ~ , &c. 
fi ( r i/ fi ( f */* [\( r ~) [i\. r 6 ) 

the sum of the errors taken with the sign + will be greater than when 

c 
= HW- 

c (r) 

Now diminish y by the quantity 5 y so that r-^j 5 y may be comprised 

c (r) C ^ + 1) 

between 1-7- and T-T- rr, the sum of the negative terms of the series (P) 
n w h ^ t */ 

will increase, in changing their sign, by the quantity 

{h 1 ) + h( 2 > + ____ h< r >} 3y; 

and the sum of the positive terms of the same series will decrease by the 
quantity 

Jh( r + 1 ) + h< r + 2 ) + &c.} ay; 
and since we have 

h) + hW + ____ h > h^ r + J ) + h( +2) + &c., 

it is clear that the entire sum of the errors, taken with the sign +, will be 
augmented. In the same manner we shall see that, by diminishing y, sp 



that it should be between . (r 1} and r- ( f-^T) or Between . (r+it; and . ^,. 



&c. the sum of the errors taken with the sign + is greater than when 

c (0 
y = p -; this value of y is therefore that which renders this sum a 

minimum. 



320 



A COMMENTARY ON [SECT. XII. & XIII. 



The value of y gives that of z by means of the equation 
z = A P . y. 

The preceding analysis being founded on the variation of the degrees 
from the equator to the poles, being proportional to the square of the sine 
of the latitude, and this law of variation subsisting equally for gravity, it 
is clear that it applies also to observations upon the length of the seconds 
pendulum. 

The practical application of the preceding theory will fully establish its 
soundness and utility. For this purpose, ample scope is afforded by the 
actual admeasurements of arcs on the earth s surface, which have been 
made at different times and in different countries. Tabulated below you 
have such results as are most to be depended on for care in the observa 
tions, and for accuracy in the calculations. 



Latitudes. 


Lengths of Degrees. 


Where made. 


By whom made. 


o.oooo 

37 .0093 
43 .5556 
47 .7963 
51 .3327 
53 .0926 
73 .7037 


25538 R .85 
25666.65 
25599.60 
25640 .55 
25658 .28 
25683 .30 
25832 .25 


Peru. 
Cape of Good Hope 
Pennsylvania. 
Italy. 
France. 
Austria. 
Laponia. 


Bouguer. 
La Caille. 
Mason & Dixon. 
Boscovich & le Maire. 
Delambre & Mechain. 
Liesganig. 
Clairaut, &c. 






SUPPLEMENT 



TO 



BOOK III. 



FIGURE OF THE EARTH. 



585. IF a fluid body had no motion about its axis, and all its parts were 
at rest, it would put on the form of a sphere ; for the pressures on all the 
columns of fluid upon the central particle would not be equal unless they 
were of the same length. If the earth be supposed to be a fluid body, 
and to revolve round its axis, each particle, besides its gravity, will be 
urged by a centrifugal force, by which it will have a tendency to recede 
from the axis. On this account, Sir Isaac Newton concluded that the 
earth must put on a spheroidical form, the polar diameter being the 




shortest. Let P E Q represent a section of the earth, P p the axis, E Q 
the equator, .(b m) the centrifugal force of a part revolving at (b). This 
force is resolved into (b n), (n m), of which (b n) draws fluid from (b) 
to Q, and therefore tends to diminish P O, and increases E Q. 

It must first be considered what will be the form of the curve P E p, 
and then the ratio of P O : G O may be obtained. 

VOL. II. X 



322 



A COMMENTARY ON 



[BOOK III. 




586. LEMMA, Let E A Q, e a q, be similar and concentric ellipses, of 
which the interior is touched at the extremity of the minor axis by P a L ; 
draw a f, a g, making any equal angle with a C ; draw P F and P G re 
spectively parallel to a f, a g ; then will P F + P G = a f + a g. 
For draw P K, Fk perpendicular to E Q, and F H, k r perpendicular to 
P K, .-. F E = E K, .-. H D = D r and PD = D K, .-. PH = Kr; 
also F H = K r, .-. if K k be joined, K k = P F; draw the diameter 
M C z bisecting K k, G P, a g, in (m), (s), (z). 
Then 

Km:Kn::Ps:Pn::az:aC::ag:ab. 
.-. K m + Ps:Kn-fPn::ag:ab 
but 

Kn + n P=K P = 2 PD = 2aC = ab.\ Km + Ps = ag. 
.. 2 Km+ 2Pszz2ag, or P F+P G = a g + a f. 
COR. PH + PI=2ai. For 

PF:PH::PG:PI::ag:ai. 
. . P F + PG:PH + PI::ag:ai::2ag:2ai. 
but 

PF + PG = 2ag, .-. PH+ P I = 2 a i. 



BOOK III.} NEWTON S PRINCIPIA. 323 




587. The attraction of a particle A towards any pyramid, the area of 
whose base is indefinitely small, cc length, the angle A being given, and 

the attraction to each particle varying as --.. 5 . 

For let 

a = area (v x z w) 

m = (A z) 

x = (A a) 

rr-i ... , section vxzw.(Aa) 2 ax 2 
Then section a b = r~ =- 



attraction = 
attraction = 



(A z) 2 * m 

a x 2 x a 



m * x l tn - 
a x 
m 1 

.*. attractions of particles at vertices of similar pyramids cc lengths. 

588. If two particles be similarly situated in respect to two similar solids, 
the attraction to the solids a lengths of solids. 

For if the two solids be divided into similar pyramids, having the par 
ticles in the vertices, the attractions to all the corresponding pyramids 
<x their lengths cc lengths of solids, since the pyramids being similarly 
situated in the two similar solids, their lengths must be as the lengths of 
the solids : .. whole attractions a lengths of the solids, or as any two 
lines similarly situated in them. 

COR. 1. Attraction of (a) to the spheroid a qf: attraction of A to 
A Q F : : a C : A C. 

COR. 2. The gravitation of two particles P and p in one diameter P C are 
proportional to their distances from the center. For the gravitation of (p) 
is the same as if all the matter between the surfaces A Q E, a q e, were 
taken away (Sect. XIII. Prop. XCI. Cor. 3.) . . P and p are similarly si 
tuated in similar solids, . . attractions on P and p are proportional to 
P C and p C, lines similarly situated in similar solids. 

589. All particles equally distant from E Q gravitate towards E Q with 

equal forces. 

X2 



324 



A COMMENTARY ON 



[BOOK III. 



For P G and P F may be considered as the axes of two very slender 
pyramids, contained between the plane of the figure and another plane, 
making a very small angle with it. In the same manner we may conceive 
of (a f ) and (a g). Now the gravity of P to these pyramids is as 
P F + P G ; and in the direction P d is as P H + PI. Again, the 
gravity of (a) to the pyramids (a f ), (a g) is as (a f + a g), or in the di 
rection (a i) as 2 a i ; but PH+PI = 2ai:.\ gravity of P in the di 
rection P d = gravity of (a) in the same direction. 

It is evident, by carrying the ordinate (f g) along the diameter from (b) 
to (a) ; the lines (a f ), (a g) will diverge from (a b), and the pyramids of 
which these lines are the axes, will compose the whole surface of the in 
terior ellipse. The pyramids, of which P F, P G are the axes, will, in 
like manner, compose the surface of the exterior ellipse, and this is true 
for every section of the spheroid passing through P m. Hence the at 
traction of P to the spheroid P A Q in the direction P d equals the at 
traction of (a) to the spheroid (p a q) in the same direction. 

590. Attraction of P in the direction P D : attraction of A in the same 
direction : : P D : A C. 

For the attraction of (a) in the direction P D : attraction of A in the 
same direction : : P D : A C, and the attraction of (a) = attraction of P. 
.*. attraction of P : attraction of A : : P D : A C. 

Similarly, the attraction of P in the direction E C : attraction of A in 
the direction E C : : P a : E C. 




591. Draw M G perpendicular to the ellipse at M, and with the radius 
O P describe the arc P n. 
Then Q G : Q M : : Q M : Q T 

Q G - Q M * 

<** = -- 



BOOK III.] 
And 



NEWTON S PRINCIPLE 

O Q : O P : : O P : O T 
OP 2 



325 



. OQ = 



. Q G : Q O : 



OT * 

QM 2 OP 2 QM 2 .OT 



Q T O T 



QT 



:OP 



but 



OT: OQ:: OP 2 : QO 2 

OT: TQ :: OP 2 : O P 2 OQ 2 

:: OP 2 : nQ 2 

:: OP 2 : PQ. Qp:: OE 
OT O E 2 



QM 



TQ ~ QM 2 * 
.-. QG: QO:: O E 2 : O P 2 

or QE2 no 

. . Q Or = ^j-p^ . Q O. 

592. A fluid body will preserve its figure if the direction of its gravity, at 
every point, be perpendicular to its surface ; for then gravity cannot put its 
surface in motion. 

593. If the particles of a homogeneous fluid attract each other with forces 

varying as jr , and it revolve round an axis, it will put on the form 

of a spheroid. 




For if P E p P be a fluid, P p the axis round which it revolves, then 
may the spheroid revolve in such a time that the centrifugal force of any 
particle M combined with its gravity, may make this whole force act per 
pendicularly to the surface. For let E = attraction at the equator, 
P = attraction at the pole, F = centrifugal force at the equator. 

X3 



326 A COMMENTARY ON [BOOK III 

Then (590), 

attraction of M in the direction M R : P : : Q O : P O 

/. attraction of M in the direction M R = 



. 

Similarly, the attraction of M in the direction M Q = E- *\ R , 

O E 

But the centrifugal force of bodies revolving in equal times oc radii. 



V 

F QC - X 



r r . P 2 

4ff 2 r 

(and P being given) cc r 

F O T? 

. . centrifugal force of M = 7^-^, 

O E 

.*. whole force of M in the direction M O = - ~ ^ . 

o hi 

p /~\ f~\ /"P "p\ f\ ~o 

Take M r = -?<- , M g = 1 , , complete the paral 
lelogram, and M q will be the compound force; O E and O P .-. must 
have such a ratio to each other that M q may be always perpendicular to 
the curve. Suppose M q perpendicular to the curve, then, by similar 
triangles, q g or M r : M g : : Q G : Q M. 

. p - Q (E F) O R . . OE 2 Q < 
P O O E : O P 2 

u^rk^i} O R O E 2 ^ ^ 

= (E * ) . TY-FT . rT&t y J 



.-. P : E F : : O E : O P, 

in which no lines are concerned except the two axes ; .*. to a spheroid 
having two axes in such a ratio, the whole force will, at every point, be 
perpendicular to the surface, and .*. the fluid will be at rest. 

P 1VT R 
59*4. The attraction of any point M in the direction M R = ; 

/. if P be represented by P O, M R will represent the attraction of M in 
the direction M R, and M v will represent the whole attraction acting 
perpendicularly to the surface. 



BOOK III.] 



NEWTON S PRINCIPIA. 




327 



Draw (v c) perpendicular to M O. 
Then 

M O: M a: : M v: M c: : attraction in the direction M v : MO. 



.. attraction in the direction M O = 



Mv.Ma OP 2 



cc 



MO "MO ~ M O * 

By similar triangles T O y, M v R, (the angle T O y being equal to the 
angle v M R.) 

T O : O y : : v M : M R 
.-. TO.MR = Oy.vM = Ma.Mv = TO.OF = OP 2 . 

595. Required the attraction of an oblong spheroid on a particle placed 
at the extremity of the major axis, the excentricity being very small. 

Let axis major : axis minor : : 1 : 1 n. Attraction of the circle 
N n (Prop XC.) 

_EL x 

EN C " Vn 2 + (1 n) 2 (2n n ) 

a 1 x [2 x n. (4 x ssn jj * 

a 1 x {(2 x)~^ + i-(2x)~*n. (4 n 2n 2 )} 



<c 



/.A ax 



V 2 4 V 2 

x i x n - 1 s 

= 2L :-=. (4x 8 x 2x* 

V 2 4, V 2 

V~2 f_ _n /8 x^ 4 x 
~^ * /i. */"> \ 3 



X 4 



328 A COMMENTARY ON [BOOK III. 

Let x = 2 E O = 2, 



/.Ax 2 



* 16 



3 4 V 2 i " * 3 5 

2_ 8ji 4 n 

.-. attraction of the oblong spheroid on E : attraction of a circum 
scribed sphere on E : : (since in the sphere n = 0.) 



596. Required the attraction of an oblate spheroid on a particle placed 
at the extremity of the minor axis. 

Let axis minor : axis major : : 1 : 1 -f- n. 

.*. A 7 cc x { 1 x \ 

V x 2 + (1 + n) 2 . (2x x 2 )J 

- ..._ _^ I 

V 2 x + 4nx 2 nx*J 

)~*4nx 2nx 2 )} 



*-- I(S 

2 ( " 

i 

. x * x n x 2 x n x 
ax -=, + 



V 2 V 2 2 V 2 

V~2 . x s V~2 . n x^ n x 
,\A cc x 



3 3 5V2 

. . whole attraction 

4 4n 4n 2,8n 4n 

cc 2 -I- cc 4- cc 1 4- 

3 3 5 3 15 5 

.. attraction of the oblate sphere on P : attraction of the sphere in- 

A. r\ 

scribed on P : : 1 -j- - : 1. 
5 

Since these spheroids, by hypothesis, approximate to spheres, they may, 
without sensible error, be assumed for spheres, and their attractions will be 
nearly proportional to their quantities of matter. But oblong sphere 
: oblate : : oblate : circumscribed sphere. . . A of oblong sphere on E : A 
of oblate on E : : A : A" of circumscribed sphere on E. 

.-.A : A"::A:A :: V~A : V~A" : . J I ~: I : : 1 ?-~: I 



BOOK III] NEWTON S PRINCIPIA. 329 

Also 

A, n 

att n . of oblate sph. on P : att n . of insc d . sph. on P : : 1 + : 1 

o 

att n . of insc d . sph. on P : att n .ofcircumsc d . sph. onE: : 1 : 1 + n 

att a . of circumsc d . sph. onE : attr n . of oblate sph. on E : : 1 : 1 - n 



.. attraction of the oblate sphere on P : attraction of the oblate sphere 

. 4 n - - 2 n 
on E : : 1 -\ =- : 1 + n . 1 

*) O 

. 4 n 3 n , n , 

::! + :!+ : : 1 + I nearly. 






n 
5" 

ri^ 3 n 2 
5T "25" 
3_n 2 
25 



.-. P : E : : 1 + ^- : 1 5 

but (593), P:E F::OE:OP 

::l + n:l::P + F:E nearly 
rli.E F = P 



.-. 1 + n. E F nF=P 
/.rr^.E nF= P+ F 
and since (n) is very small, as also F compared with E, 

.-. r+ir. E = P + F 

.-. 1 + n : 1 : : P + F : E 



E + ^_+ F:E:: 



5 

_ 4 n E 
f . -g 

5 F 

n = 4E 
4 E : 5 F : : 1 : n 



380 A COMMENTARY ON [BOOK III. 

or " four times the primitive gravity at the equator : five times the centri 
fugal force at the equator : : one half polar axis : excentricity." 

597. The centrifugal force opposed to gravity a cos. 2 latitude. 




Q 



o 



E 



Let (m n) = centrifugal force at (m), F = centrifugal force at E. 
.*. (n r) is that part of the centrifugal force at (m) which is opposed to 
gravity. 

Now -\ 

F: mn:: O E: Km /. . F : n r : : O m 2 : K m 2 

and ( : : r 2 : cos. 2 lat. 

mn:nr ::Om:Km j 

. . m r oc cos. 2 lat. 

598. From the equator to the pole, the increase of the length of a de 
gree of the meridian cc sin. 2 lat. 




Q C E 

nr:Ms::nG:MG::CP:CR::l n:J. 
.. n r = 1 n . M S = 1 n . <p f sin. d = 1 n . cos. 6 . 

m r = s t = <p . cos. 6 = sin. 6 . tf 
.. m r 2 = sin. z 6 . 6 * 

.-. mn 2 = nr 2 + m r 2 = tf*. sin. 8 6 + (1 n) 1 . cos. O. tf* 
= 6 *. (sin. 2 6 + 1 2 n . cos. 2 ^) 
= 6 z (sin. * 6 + cos. 2 6 2 n . cos. 2 6) 
tf*. (1 2 n. cos. 1 6) 
.-. m n = tf. (I n . cos. * 6) 
.. at the equator, since 

6 = 0; m n = ff (1 n) 



BOOK III.] NEWTON S PRINCIPIA. 331 

.. increase = 6 ( 1 n . cos. 2 d 1 + n) = 6 . n ( 1 cos. * 6) 

= V. n sin. * 6, 
.\ increase n ff. sin. z 

sin. 2 0, cc sin. z latitude. 

599. Given the lengths of a degree at two given latitudes, required the 
ratio between the polar and equatorial diameters. 

Let P and p be the lengths of a degree at the pole and equator, m and 
n the lengths in latitudes whose sines are S and s, and cosines C and c. 
Then as length of a degree oo radius of curvature, (for the arc of the me 
ridian intercepted between an angle of one degree, which is called the 
length of a degree, may be supposed to coincide with the circle of curva 
ture for that degree, and will .. cc radius of curvature.) 

CD 2 
.Y _ 

PF 

Now at the pole CD* becomes AC 2 , and P F becomes B C 

.*. length of a degree cc . ; oc ^- ; 

Jti v> D 

similarly the length of a degree at the equator 

B C 2 b 2 

k xc~ >oc r* 

P : p : : ~ : : : a 3 : b 3 : : 1 : (1 n ) 3 . 
b a 

Now 

m p : n p.(59S):: S 2 : s 2 , 

.-. m n : n p : : S 2 s 2 : S 2 , 

m n.S 2 
n -P= 82 _ s * * 

but 



T n m n.S 

P_ p : n p :: i 2 : s 2 :: P p : 



.-. P - p = 




S 2 s 



332 



A COMMENTARY ON 



[BOOK III. 



n S 2 n s 2 m s 



n S 



ms~ 



m 
= P + 02 



S 2 s 2 
n n S 



S 2 s 2 : 
m s 2 + m n 



S 2 s 2 

m.(l s 2 ) n.(l 

S 2 s 2 
mc 2 -~n C e 



... P : p 



me* n C 

S 2 s s 



S 2 
n S 2 m s 



S 2 s 2 

:: me 2 n C 2 : n S 2 m s 2 :: 1 : (1 n ) 3 
.-. (m c 2 n C 2 ) I : (n S 2 m s 2 ) J : : 1 : 1 n . 

600. The variation in the length of a pendulum oc sin. 2 latitude. 
Let 1 = length of a pendulum vibrating seconds at the equator. 

L = length of one vibrating seconds at latitude &. 
The force of gravity at the pole = 1, .-. the force of gravity at the equator 
= 1 F, and the force of gravity in latitude 6 (603) = 1 F. cos. 2 6, 

.-. L : 1 : : 1 F. cos. 2 d : 1 F (since a a a F) 
.-. L 1 : 1 : : F. ( 1 cos. 2 6) : 1 F : : F. sin. 2 6 : 1 F, 

1 F. sin* 



.-. L 1 = 



oc sin. 2 0. 



1 -F 

From the poles to the equator, the decrease of the length of a pendu 
lum always vibrating in the same time, oc cos. z latitude. 

Let L = length of a pendulum vibrating seconds at the pole, 
.-. L : L :: 1 : 1 F. cos 2 *?, 
... L : L L :: 1 : F. cos 8 *, 
.. L 7 L cos. 2 6. 

601. The increase of attraction from the equator to the pole oc sin. 2 lat. 

Let 

O E : O P : : 1 : 1 n. 

Let 

M O = a, the angle M O E = 

PO 2 
^~* C y^V T~l 



.-.MR 2 = 



O E 



OR 2 }, 



or 




a*.sin. 8 = (1 n) 2 . (1 a* cos. 2 6) 
= l~^-2n. (1 a cos. 2 6) 
. . a 2 . { sin. 2 6 + 12 n. cos. 6} = 1 2 n 



BOOK III.] NEWTON S PRINCIP1A. 333 

B _ _ 1 2 n _ _ 1 2 n 

* a ~" sin. 2 d + cos. 2 & 2 n. cos. 2 6 ~ 1 2 n. cos. 2 O 9 

1 n ^ - 1 + n. cos. *& - - . ,. 

.-. a = ^ - =-. = 1 n-TF^ i -- A = 1 n.(l+ncos. 2 0), 
1 n. cos. 2 d 1 2 n 2 . cos. 4 6 

i_ n (l cos. 2 g) = 1 n. sin. 2 0, 

- = i - : n = 1 + n . sin. 2 6 = ^-^ , 
a 1 n . sin. 2 d MO 

but (594) the attraction in the direction M O oc ^F-Q , 

.-. attraction in the direction M O (A) : attraction at E (A ) 

: : 1 + n . sin. 2 6 : 1, 
.-. A A : A : : n . sin. 2 6 : 1, 
.-. A A = A , n . sin. ~ &, 
.*. increase of attraction oc sin. 2 d oc sin. 2 latitude. 

602. Given the lengths of two pendulums vibrating seconds in two 
known latitudes ; find the lengths of pendulums that will vibrate seconds 
at the equator and pole. 

Let L, 1 be the lengths of pendulums vibrating seconds at the equator 
and pole. 

L , 1 be the lengths in given latitudes whose sines are S, s, cosines C, c. 

.-. L L : 1 L : : S 2 : s 2 
.-. L s 2 Ls 2 = F S 2 Ls 2 
.-. L. (S 2 s 2 ) = 1 S 2 L s 8 , 
1 S 2 L s 2 
S s~ 
Again 

J-/ Lt , 1 LI , I o ! Ij 

.-. L L = 1 S 2 LS 2 , 

L L. (1 S 2 ) 

S* 

L/ (F S 2 L s 2 )(l S 2 ) 
S 2 S 2 . (S 2 s 2 ) 

U S 2 L x s g T S* + V S 4 + L X s 2 L S 2 s c 



S 2 . (S 2 s 2 ) 

L / S 2 V. S 2 + I . S 4 - L . S 2 s 2 

S 2 . (S 2 s 2 ) 

L . (1 s 2 ) l .(l S 2 ) Uc 8 I 7 C 

- S 2 s x ~ S 2 s 2 



334 A COMMENTARY ON [Boon III. 

603. Given the lengths of two pendulums vibrating seconds in two 
known latitudes; required the ratio between the equatorial and polar 
diameters. 

Since the lengths oc forces, the times being the same, 
. . L : 1 : : force at the equator : force at the pole 

: : ( 10 ) 7 : T~7{ :: 1 n:l::OP:OE, 
. . O P : O E : : polar diameter : equatorial diameter 

: : L : 1 : : I S z L s 2 : L c 8 1 C 2 . 

604. To compare the space described in one second by the force of gra 
vity in any given latitude, with that which would be described in the same 
time, if the earth did not revolve round its axis. 

The space which would be described by a body, if the rotatory motion 
of the earth were to cease, equals the space actually described by a 
body at the pole in the same time ; and if the force at the pole equal 1, 
the force at the latitude 6 (597) equal 1 F . cos. 2 0, and since S = m F T 2 , 
and T is the same, .-. S F. 

.*. space actually described when the earth revolves : space which 
would be described if the earth were at rest : : 1 F. cos. 2 6 : 1. 

605. Let the earth be supposed a sphere of a given magnitude, and to re 
volve round its axis in a given time ; to compare the weight of a body 
at the equator, with its weight in a given latitude. 

V- 4 T 2 . r 
The centrifugal force = = ~ = F equal a given quantity, 

since (r) and P are known. Now the force at the equator =1 F, 
and the force at latitude 6 = 1 F . cos. 2 d, and the weight attractive 
force 

.-. W : W :: 1 F : 1 F.cos. 2 0. 

606. Find the ratio of the times of oscillation of a pendulum at the 
equator and at the pole, supposing the earth to be a sphere, and to re 
volve round its axis in a given time. 

L oc F T 2 but L is constant, .-. T 2 -_ , 

/. T. oscillation at the pole : T. oscillation at the equator 
: : V force at the equator : V force at the pole 



BOOK III.] 



NEWTON S PRINCIPIA. 



335 



607. If a spherical body at rest be acted upon by some other body, it 
may put on the form of a spheroid. 




Let P E p be the earth at rest; (S) a body acting upon it; (O) its cen 
ter; (M) a particle on its surface. 
Let P = polar, ") 

E = equatorial,/ attraCtl n n the earth 

Then the attraction on M is parallel to M Q = 



E. OR 
OE 



Similarly the attraction on M is parallel to M R = ^ . 

Let (m) =: mean addititious force of S on P. 
(n) = mean addititious force of S on E. 
Now since the addititious force (Sect. XL) a distance, 

m. M O 



.-. the whole addititious force of S on M = 



PO 



and 



- ~ : addititious force in the direction M R : : M O : MR, 



.-. addititious force in the direction M R = 

Again, since 

m : n : : P O : E O, 
m n 



m t 



.-. whole addititious force of S on M = " ]V 1 , 

EJ O 



330 A COMMENTARY ON [Boo* III. 

.-. addititious force in the direction M Q = n -^ Q = - n ^ - , 

lii O 1 j O 

.-. whole disturbing force of S on M in the direction M Q twice the 

2 n OR. 

addititious force in that direction, and is negative = ^- . 

. . whole attraction of M in the direction M Q = [E 2 n}. ~r4 . > 

and the whole attraction of M in the direction M R = {P + m}. 

w x 

Take M g = [E - 2 n} . |^ 

Mr = {P + mj . ~|{ 

complete the parallelogram (m q), and produce M q to meet P p in G. 

Now if the surface be at rest, M G will be perpendicular to the sur 
face. 

. . M r : M g : : g q : g M : : G Q : Q M, 






or 



.-. P + rn : E 2 n : : O E : O P. 
.*. figure may be an ellipse. 

608. Suppose the Moon to move in the equator ; to find the greatest ele 
vation of tide. 



A n 



Let A B C D be the undisturbed 
sphere; M P m K a spheroid 
formed by the attraction of the 
Moon; M the place to which the 
Moon is vertical. 

Let 

(A E = i 

<EM = 1 - 

(E F = i , _ 

Then since the sphere and spheroid have the same solid content, 

. 4 ^r. (A E) 3 _4-r.EM.(FE) t 
3 3 




BOOK III.] NEWTON S PRINCIPIA. 337 

.-.1 = 1 + _ 2 /3 2/3 + /3 2 + /3 2 

= 1 + 2/3 nearly, () and (|6) being very small, 
.-.a = 2 j3 or greatest elevation rr 2 X greatest depression. 
614. To find the greatest height of the tide at any place, as (n) . 
Let 

E P = <? z. P E M = tf + /3 = 3 ^ = EM E F = M, 

.-. PN 2 = 8 .sin. 8 tf = . {EM 2 EN 2 ? 



/ J _ Q \ 2 

Now 7^ - --(- by actual division (all the terms of two or more dimen- 
(1 + a) 2 

sions being neglected) = 1 2 . (a + /3) = 1 2M, 

.-. PN 2 = g 2 . sin. M = (1 2 M). [1 + 2 a 2 . cos. 2 ^ 

(since 2 a = L? i = 4-M) = (1 2M) U + ^ f 2 . cos.^J. 
/& o o o 

/ 4 M 

.-. s *. [sin. M + (1 2M).cos. 2 ^ = (1 2 M). 



2M 2 M 



sin. 2 tf + cos. z 6 2 M . cos. 2 6 1 2 M . cos. 2 0, 



-T 



= i + M . cos. i l ; 

9 

M 

.-. g 1 = M . cos. 2 tf --- -- =EP En = Pn= elevation re- 

8 

quired. 

M 

615. Similarly if the angle M E p = tf, .-. E p = 1 + M cos. 2 ff ~ , 

B 

M 

.\1 E p = p n = depression = -^ -- M . cos. 2 6 

O 

9 TVT 9 M 

= M M . cos. 2 tf ^ = M sin. .f 44 ifS . 

o o 

9 ivr 

616. B M = a = =-j=, 

o 

.-. BM-Pn=^+ M.sin.M ~ 
o o 

= M . sin. 2 d QC sin. 2 6, 
VOL. II. Y 



338 A COMMENTARY ON [BooK III. 

.*. greatest elevation oc sin. 2 horizontal angle from the time of high tide. 
617 At (O) Pn = 0, 

M 

.-. M . cos. 2 6 = 0, 

O 

.-. M . cos. *0 = ~ , 

. . cos. 6 = =^. 
VB 

. . 6 = 54 , 44 . 

Hitherto we have considered the moon only as acting on the spheroid. 
Now let the sun also act, and let the elevation be considered as that pro 
duced by the joint action of the sun and moon in their different positions. 

Let us suppose a spheroid to be formed by the action of the sun, whose 
semi-axis major = (1 + a), axis minor = (1 b). 

618. Let (a + b) = S, (<p) = the angular distance of any place from the 
point to which the sun is vertical. It may be shown in the same manner 
as was proved in the case of the moon, that 



and 



S 

S . cos. 2 <f> ~- =: elevation due to the sun, 

o 

2 S 
S . sin. 2 <ff ~^~ = depression due to the sun, 

9 



(<p } being the angular distance of the place of low water from the point to 
which the sun is vertical, 

.. M . cos. 2 6 + S . cos. 2 <p = compound elevation. 

o 



Similarly M . sin. 2 6 + S . sin. 2 <f> f M + S = compound depres 
sion. 
610. Let the sun and moon be both vertical to the same place, 

/. 6 = <? = 0, 

AT I O O 

... M + S "^ =-J-M + S = compound elevation, 
3 o 

and 

6 = ? = 90, 



.-. M + S f.M+!S=^M+S = compound depression, 
. . compound elevation + compound depression = M -f- S = height of 
spring tide. 

620. Let the moon be in the quadratures with the sun, then at a place 
under the moon, 

(6) = 0, and (9) = 90, 



BOOK III.] 



.. compound elevation = M 



NEWTON S PRINCIPIA. 
M+ S 



339 



also 



(6") = 90, and (?) - 0, 




.. compound depression = M f . M -f- S, 
.-. height of the tide at the place under the moon = 2 M M + S 

= M -f- S = height of neap tide. 
Similarly at a place under the sun, height of tide = S M. 

621. Given the elongation of the sun and moon, to find the place of com 
pound high tide. 

Compound elevation = M cos. * & -f- S 

M + S 
cos. 2 <p ^ = maximum at high 

water. 

.-. 2 M cos. sin. & 6f 2 S 
cos. <p sin. <p <p = 0, 
but 

(6 + <p) = elongation = JE 

= constant quantity, 
.-. ff + f = 
.-. (f = - f, 
.-. 2 M cos. 6 sin. = 2 S cos. <p sin. <p , 
.-. M sin. 2 6 S sin. 2 <p, 

.-. M : S : : sin, 2 <p : sin. 2 0, 
.-. M + S : M S : : sin. 2 <f> + sin. 2 6 : sin. 2 <p sin. 2 tf, 

: : tan. (<p + 6) : tan. (<p 6} t 

and since (<p + 6) is known, .-. (<p 6} is obtained, and . (p) and (0) are 
found, i. e. the distance of the sun and moon from the place of compound 
high tide is determined. 

622. Let P be the place of high tide, 

P the place of low water, 90 distant from P, 

Pm = Pml = 90 + = / Ps = p P s 

= 90 <p = <p f . 

Now the greatest depression = M sin. 2 & + S sin. 2 <p f M + S, 
but 

sin. 8 6 = sin. * (90 + 6) = sin. 2 supplemental angle (90 6) = cos. 2 4, 
and 

sin. 2 <p f = sin. * (90 <p) = cos. 2 p, 

.-. the greatest depression M cos. 2 6 + S cos. * <p f M + S, 
and the greatest elevation = M cos. * 6 + S cos. * <p M + S, 
.-. the greatest whole tide = the greatest elevation + greatest depression 

i 4/ 



340 



A COMMENTARY ON 

= 2 M cos. * + 2 S cos. * <p NT+~$~, 



[BOOK III. 



= M [2 cos. * 61} + S (2 cos. 2 p 1) 
= M cos. 2 + S cos. 2 p. 
623. Hence Robison s construction. 

A 




Let A B D S be a great circle, S and M the places to which the sun 
and moon are vertical ; on S C, as diameter, describe a circle, bisect S C 
in (d); and take S d : d a : : M : S. Take the angle S C M = (<p + 6), 
and let C M cut the inner circle in (m), join (m a) and draw (h d) par 
allel to it; through (h) draw C h H meeting the outer circle in H; then 
will H be the place of high water. 

For draw (d p) perpendicular to (m a) and join (m d). 

Let the angle S C H = p, and the angle M C H = d. 

Since M : S 
.-. M + S: M S 



Sd:da 

Sd + da:Sd da 

d m + da:dm da 

d a m + d m a 



t an . 



tan. 



Sdm 



tan. 

dam dma 
tan. -- ^ - 

S d h m d h 



dam d m a 



tan. S C M : tan. 

tan. S C M : tan. (S C H H C M) 
tan. (p + 6) : tan. (<p 6) 
. . H is the place of high water 621. 
Also (m a) equals the height of the whole tide. For (a p) = a d. cos. pad 

= S. cos. S d h = S. cos. 2 <p 
and 

(p m) = m d. cos. p m d = M. cos. m d h = M. cos. 2 6 



BOOK III.] 



NEWTON S PRINCIPIA. 



341 



.-. a m = a p + p m = M. cos. 2 + S. cos. 2 p = height of the tide. 
At new moon, & = <p = 1 tide _ M , g _ tide> 

At full moon, 6 = 0, p = 180 J 
When the moon is in quadratures, (m a) coincides with C A, 

.-. 6 = 0, p = 90, 
.-. tide = M S = neap tide. 

624. The fluxion of the tide, i. e. the increase or decrease in the height 
of the tide a p . (m a) oc p . {M. cos. 2 6 + S. cos. 2 ?}. But the sun 
for any place is considered as constant, 

.-. <p . (m a) oc M. sin. 2 6. 2 6 , 
.-. <p . (m a) is a maximum at the octants of the tide with the moon 

ex M. sin, 2 
since at the octants, 2 6 = 90. 

The fluxion of the tide is represented in the figure by (d p). 

For let (m u) be a given arc of the moon s synodical motion, draw (n v) 
perpendicular on (m a), .*. (m v) is the difference of the tides. 

Now mu:mv::md:dp and m u and m d are constant, .-. 
m v d p and d p is a maximum, when it coincides with (d a), i. e. when 
the tide is in octants; for then 2 (m a d) = 90. 

625. At the new and full moon, it is high water when the sun and 



M 




moon are on the meridian ; i. e. at noon and midnight. At the quadra 
tures of the moon, it is high water when the moon is on the meridian, 
because then (m) coincides with C. 

For let M. cos. * d + S. cos. 2 p - = maximum; then since 

in quadratures (p + 6) = 90, . . P = 90 6, 

... M. cos. * 6 + S. sin. 2 6 3 M + S = maximum, 

. . 2 M. cos. 6. sin. 6. ff = 2 S. sin. 6. cos. 6. 6 , 
... M S . 2 . sin. 4. cos. 6 = M S . sin. 20=0, .. sin. 2 = 0, 
.-. & = 0, that is, the moon is on the meridian. 
Y3 



34-2 



A COMMENTARY ON 



[BOOK III. 



626. From the new moon to the quadratures, the place of M16 , 
tide follows the moon, i. e. is westward of it ; since the moon moves 
from west to east, from the quadratures to the full moon, the place of 
high tide is before the moon. There is therefore some place at which its 
distance from the moon (6} equals a maximum. 

Now (621) M : S : : sin. 2 <p : sin. 2 d 

.-. M. sin. 2 6 = S. sin. 2 <f> 
.-. M. 2 V. cos. 2 6 = 8. 2 p . cos. 2 
. . cos. 2 <p = 0, .-. <p = 45. 

627. By (621) M. sin. 2 6 = S. sin. 2 <p 

. . V. M . cos. 26= <p . S . cos. 2 <p 
but 

<p + 6 = e, . . <f> -f- ^ == e , 
.-. ( e <ff) M . cos. 2 4 = <f/. S . cos. 2 f 

.: e . M . cos. 2 6 = <p f . {S. cos. 2 p -f M . cos. 2 6} 

e . M . cos. 2 

~ M . cos. 2 -f S . cos. 2 p 

Next, considering the moon to be out of the equator, its action on the 
tides will be affected by its declination, and the action of the sun will not 
be considered. 

M 

By Art. (614) the elevation = M cos. 2 6 ~ 

o 

.*. elevation above low water mark = M . cos. 2 6 -f b 

3 

now 

M 

= 2 = ! 3 

.-. elevation above low water = M . cos. * d 

=. magnitude of the tide. 

Let the angle Z P M which measures the time from the moon s pass 
ing the meridian equal t. a Z 
Let the latitude of the place 

QftO P 7 1 I7i . 

r L = 1 E / M 

Let the declination 
= 90 P M = d 



cos. ZPM = cos - ZM-cos. Z Pcos. P M 



or 



cos. t = 



sin. Z P sin. Z M 
cos. 6 sin. 1 sin. d 



cos. 1 cos. d 
. cos. 6 = cos. t cos. 1 cos. d -j- sin. 1 sin. d 




Q 



BOOK III.j NEWTON S PRINCIPIA. 343 

.-. magnitude of the tide = M. {cos. t cos. 1 cos d + sin. 1 sin. d] 2 
.-. for the same place and the same declination of the moon, the magni 
tude of the tide depends upon the value of (cos. t). Now the greatest 
and least values of (cos. t) are (+1) and ( 1), and since the moon only 
acts, it is high water when the moon is on the meridian, and the mean 

greatest -f least 
tide = i__X -, 

greatest = M. { sin. 1 sin. d + cos. 1 cos. d} 2 
least = M. {sin. 1 sin. d cos. 1 cos. d} 2 
... Shiest- + least = M ^.^ , j ^ a d + ^ , } ^ 2 d} 

4 

2 sin. 2 1 = 1 cos. 2 1 
2 sin. 2 d = 1 cos. 2 d 

.-. 4. sin. 2 1 sin. 2 d = 1 {cos. 2 1 + cos. 2 d} + cos. 2 1 cos. 2 d 
2. cos. 2 1 = cos. 21 + 1 
2. cos. z d = cos. 2 d + 1 

.-. 4. cos. 2 1 cos. 2 d = 1 + (cos. 2 1 + cos. 2 d) + cos. 2 1 cos. 2 d 
.-. 4. {sin. 2 1 sin. 2 d + cos. 2 1 cos. z d} = 2 + 2. cos. 2 1 cos. 2 d 
.-. mean tide = M. sin. 2 1 sin. 2 d + cos. z 1 cos. * d 

M * + cos - 2 * cos> 2 d 

SB 

It is low water at that place from whose meridian the moon is distant 
90, /. cos. 6 0, /. for low water 

cos t _ _ sin - \ sin \ = - tan. 1 tan. d. 
cos. 1 cos. d 

When (1 + d) = 90, . . tan. 1 tan. d = tan. 1 tan. (90 1) 

tan. 1 

= tan. 1 cot. 1 = ; 1 

tan. 1 

. cos. t = 1, .*. t = 180, . . time from the moon s passing the meri 
dian in this case equals twelve hours, .-. under these circumstances there 
is but one tide in twenty-four hours. 

When 1 = d, .-. cos. t = tan. z 1 
and the greatest elevation = M {cos. t cos. 1 cos. d + sin. 1 sin. d} 2 

(since cos. t = 1) = M. {cos. * 1 + sin. 8 1} = M. 
When d = 0, /. greatest elevation = M cos. 8 1. 
When 1 = 0, .-. greatest elevation = M cos. 8 d. 

At high water t = 0, . . greatest elevation when the moon is in the 
meridian above the horizon, or, the superior tide = M {cos. 1 cos. d + 
sin. 1 sin. d} " = M cos. 8 (1 d) = T. 

For the inferior tide t = 180, /. cos. t = 1, 

y 4 



344 



A COMMENTARY ON 



[BOOK III. 



.-. inferior tide = M {sin. 1 sin. d cos. 1 cos. d? 2 

= M { 1 (cos. 1 cos. d sin. 1 sin. d)} 2 
= M cos. 2 (1 + d) = T . 
Hence Robison s construction. 

With C P = M, as a radius, describe a circle P Q p E representing 

P 

Z 

xV N 

M 




N 



a terrestrial meridian ; P, p, the poles of the earth ; E Q the equator ; 
(Z) the zenith; (N) the nadir of a place on this meridian; M the place 
of the moon. Then 

Z Q latitude of the place = I \ 

M Q declination = d / " Z M the Zenith distance = l ~ d - 

Join C M, cutting the inner circle in A ; draw A T parallel to E Q. 
Join C T and produce it to M ; then M is the place of the moon after 
half a revolution, .. M x N = nadir distance 

= ME + EN = MQ + ZQ = l + d. 

Join C Z cutting the inner circle in B; join B with the center O 
and produce it to D ; join AD, B T, A B, D T ; and draw T K, A F 
perpendiculars on B D. 

^ADB = ^BCA = ZQ M Q=l-d ) 

^TDB = 180_^TCB=AMCN=l+d/ andtlleangIesB A D 
B T Z are right angles 

BD:DA::DA:DF= 



B D 



D 



== .cos . - 



= M cos. (1 d) = height of the sup r . tide. 



BOOK III.] NEWTON S PRINCIPIA. 345 

Again 



= M cos. 1 + d = point of the inferior tide. 

If the moon be in the zenith, the superior tide equals the maximum. 
For then 1 d = 0, .-. cos. I d = maximum, and B D = D F. 
If the moon be in the equator, d = 0, . . D F = D K. 
The superior tide = M cos. 2 (1 d) = T 
The inferior tide = M cos. 2 (1 + d) = T. 

Now T > T , if (d) be positive, i. e. if the moon and place be both on 
the same side of the equator. 

T < T if (d) be negative, i. e. if the moon and place be on different 

sides of the equator. 

If (d) = 90 1, . .D K= Mcos. 2 (1+ 90 1) = M cos. 2 ! 
If (d) = 90 + 1, and in this case (d) be positive, and (1) negative, 
.-. D F = cos. 2 (d 1). M = M cos. 2 (90 +1 1) = M cos. ~ 90 = 0. 



PROBLEMS 



FOR 



VOLUME III. 



PROB. I. The altitude P R of the 
pole is equal to the latitude of the place. 

For Z E measures the latitude. 

= P R by taking Z P from E P and 
ZR. 

PROB. 2. One half the sum of the H 
greatest and least altitudes of a cir- 
cumpolar star is equal to the altitude of 
the pole. 

The greatest and least altitudes are at 
x, y on the meridian. 

Also 




R = 2(Py+Ry) = 2 . altitude of the pole. 

PROB. 3. One half the difference of the sun s greatest and least meridian 
altitudes is equal to the inclination of the ecliptic to the equator. 

The sun s declination is greatest at L, at which time it describes the 
parallel L r. 

. . L H is the greatest altitude, 

The sun s declination is least at C, when it describes the parallel 
sC. 

. . s H is the least altitude, 
and 

4.(LH sH) = 4 Ls = LE. 

PROB. 4. One half the sum of the sun s greatest and least meridian al 
titudes is equal to the colatitude of the place. 



= * (2 H E) = H E. 



348 



PROBLEMS 



K 




PROB. 5. The angle which the equator makes with the horizon is equal to 
the colatitude = E H. 

PROB. 6. When the sun describes 
b a in twelve hours, he will describe c a 
in six ; if on the meridian at a it be 
noon, at c it will be six o clock. Also 
at d he will be due east. He travels 15 
in one hour. The angle a P x, mea 
sured by the number of degrees con 
tained in a x (supposing x equals the 
sun s place), converted into the time at 
the rate of 15 for one hour, gives the 
time from apparent noon, or from the 
sun s arrival at a. 

PROB. 7. Given the sttn s declination, and latitude of the place ; find the 
time of rising, and azimuth at that time- 

Given Z E, .-. Z P = colat. given. 
Given be, . . P b = codec, given. 
Given b Z = 90. 

Required the angle Z P b, measuring 
a b, which measures the time from sun 
rise to noon. 

Take the angles adjacent to the side 
90, and complements of the other three 
parts, for the circular parts. 

.-. r. cos. ZPb = cot. ZPcot. Pb 
or 

r . cos. hour ^.=tan. lat. tan. dec. 

.. log. tan. lat. + log. tan. dec. 10 = log. cos. hour L. required. 

Also the angle P Z b measures b R, the azimuth referred to the north, 

and 

r . cos. P b = cos. P Z . cos. Z 

r . cos. p 

.. cos. L = = f- 1 - . 
sin. L 

PROB. 7. (a) r. cos. hour L. tan. latitude tan. declination, for sun rise. 

2 . tan. lat. tan. dec. 
Hence the length of the day 2 . cos. hour L. = 




FOR VOLUME III. 



349 



h may be found thus, from A Z P b cos. h = 
sin. L co 



cos. Z b Z cos. P. cos.P b 



= (sinceZb=90,) 



sin. Z P . sin. P b 
, or since h > 90, 



- . 
cos. L . sin. p 

cos. h = tan. L . cot. p, or cos. h = tan. L . cot. p. 
and the angle P Z b may be similarly found, 

r, cos. P b cos. Z P . cos. Z b 

r. COS. L = - : - 7jm - : - rT~\ - 

sin. Z P . sin. Z b 
cos, p 
cos. L 

PBOB. 8. Find the sun s altitude at six o clock in terms of the latitude 
and declination 

The sun is at d at six o clock. The angle Z P d = right angle. 
Z p = colat. P d = codec. Required Z d ( = coalt.) 

r . cos. Z d = cos. Z P . cos. d P 
or 

r. sin. altitude = sin. latitude X sin. declination. 

PROB. 9. Find the time when the sun comes to the prime vertical (that 

vertical whose plane is perpendicular to the meridian as well as to the. hori 

zon J, and his altitude at that time, in terms of the latitude and declination. 

Z P = colatttude. Pg = codeclination. The angle P Z g = right angle. 

Required the angle Z P g. 

.-. r . cos. Z P g = tan. Z P . cot. P g. 

= cot. latitude tan. declination. 
Also required Z g equal to the coaltitude, 

r . cos. P g = cos. P Z . cos. Z g. 

r . sin. declination . , ., , 
.. - ; , -. ; - =r sin. altitude. 
sin. latitude 

PROB. 10. Given the latitude, declina 
tion, and altitude of the sun ; Jind the 
hour and azimuth. 

Let s be the place. 

Given Z P, Z s, P s. Find the angle 
ZPs. 

Let Z P, Z s, P s = a, b, c, be given, 

to find B. 

2r 



E 



sin. B = 



sin. a . sin. c 



V s . (s a) . (s b) . (s c) 
where s = . 




350 



Also find C . V 



PROBLEMS 

. (Or by Nap. 1st and 2d Anal.) 



sin. C = 



2 r 

sin. a . sin. b * 



2 r 



Similarly, sin. A = sin. L- of position =. . r 
J sin. b. 



sin. c 

PROB. 1 1. Given the error in the altitude Find the error in the time 
in terms oj" latitude and azimuth. 

Let m n be parallel to H, and n x be 
the error in the altitude. 
.*. L. m P x = error in the time = y z. 

y z : m x : : rad. : cos. m y 

m x : x n : : rad. : sin. n m x 
. . y z : x n : : r z : cos. my. sin. n m x 



or 



y z = 



r*. n x 



cos. m y . sin. n m x 
r*. n x 



but 



cos. m y . sin. Z x P 

sin. Z x P sin. Z P 




Q 



sin. x Z P " 
.-. sin. Z x P = 



sin. P x 

sin. P Z . sin, x Z P 

cos. m y 

r 2 . n x 





" y ~ cos. L. sin. azimuth 

COR. Sin. of the azimuth is greatest when a z = 90, or when the sun 
is on the prime vertical, .*. y z is then least. 

Also, the perpendicular ascent of a body is quickest on the prime 
vertical, for if y z and the latitude be given, n x a azimuth, which 
is the greatest. 

PROB. 12. Given the latitude and 
declination. Find the time when twilight 
begins. 

(Twilight begins when the sun is 18 
below the horizon.) 

h k is parallel to H R and 18 below 
HR. 

.. Twilight begins when the sun is in 
hk. 

.-. Zs = 90 +18, Ps = D, ZP = colat 
Find the angle Z P s. 




FOR VOLUME III. 



351 




PROS. 13. Find the time when the 
apparent diurnal motion of a Jixed star 
is perpendicular to the horizon in terms of 
the latitude and declination. 

Let a b be the parallel described by 
the star. 

Draw a vertical circle touching it at 
s. 

. . s is the place where the motion ap 
pears perpendicular to H R. 

.-. Z P, P s, and L. Z S P = 90 is given. 
Find Z P s. 

PROB. 14. Find the time of the shortest twilight, in terms of the latitude 
and declination- 

a b is parallel to H R 18 below H R. 
The parallels of declination c d, h k, 
are indefinitely near each other. 

The angles v P w, s P t, measure 
the durations of twilight for c d, h k. 

Since twilight is shortest, the incre 
ment of duration is nothing. 
.-. v P w = s P t 

.. v r = w z 
and r s = t z 
and the angle v r s = right angle 

= w z t. 
.-. L. r v s = z w t, and L. Z w c = 90 z w t = 90 Z w P. 

.-. L z w t = Z w P. 
Similarly, 

z.rvs = Zv P 
.-. Z w P = Z a P. 

Take v e = 90. Join P e. Draw P y perpendicular to Z c. 
In the triangles Z P w, P v e, Z w = e v, P w = P v, and the angles 
contained are equal,-- .*. Z P = P e. 

.. In the triangles Z P y, P e y, Z P = P e, P y com. ; and the 
angles at y are right angles. 
. . Z e is bisected in y. 

r . cos. P v = cos. P y . cos. v y 
r . cos. P e = cos. P y . cos. y e. 




852 



PROBLEMS 



.. cos. P v : cos. P e : : cos. v y : cos. y e 
(but v y is greater than 90, . . therefore cos. v y is negative.) 

: : cos. ( compl. y e) : cos. y e 
: : sin. y e : cos. y e 
: : tan. y e : r- 



sin. L. tan. y e 

COS. p = - ^ rr 



T 18 

sin L> tan " 



sin. L. tan. 9 



P Z is never greater than 90, Z y is equal to 9, .. P y is never greater 
than 90, .*. cos. Py is always positive; v y is always greater than 90, 
.. cos. v y is always negative, .*. cos. P v is negative, . . the sun s decli 
nation is south. 

Also, if instead of R b = 18, we take it equal to 2 s equal the sun s 

j. , c ., sin. L. tan. s , 
diameter, we get from the expression sin. D = the time 

when the sun is the shortest time in bringing his body over the horizon. 
PROB. 15. Find the duration of the shortest twilight- 

z.wPZ = vPe, .-. z. Z P e = v P w. 
.*. 2 Z P e is equal to the duration of the shortest twilight. 

r . sin. Z y = sin. Z P . sin- Z P y 
or 

. sin. 90 . r 

sin. Z P y = = , 

cos. L. 

which doubled is equal to the duration required. 

PROB. (A). Given the sun s azimuth at six, and also the time when 
due east. Find the latitude. 
From the triangle Z P c, 
r . cos. L = tan. P c . cot. P Z c. 
From the triangle Z P d, 
r . cos- h = cot. L . cot. P d. 
cos. L 



.-. tan. P c = 
cot. P d = 

.-. tan. P d = 

cos. L 
cbtTZ " 

.-. sin. L = 



cot. Z 
cos. h 





cos. h 



FOR VOLUME III. 



353 




PROB. 16. Find the decimation when 
it is just twilight all night. 

Dec. bQ = QR bR 

= colat 18 

= 90 L 18 
= 72 L 

PROB. 17. Given the declination, 
find the latitude, the sun being due east, 
when one half the time has elapsed be 
tween his rising and noon. 

Given L Z PC, and Z P d = | Z P c. 

Given also P d = p, 
and A P Z d right angle. 

v by Nap. 

r . cos. h = tan. Z P . cot p 

. T r. cos. h 

v cot jL = . 

cot. p 

If the angle Z P c be not given. 
From the triangle Z P d, 

. cos. Z P d = tan. Z P . cot p. 

From the triangle Z P c, 

r cos. Z P c = cot. Z P . cot p, 

or cos. h = cot X. cot p^ 

cos. 2 h = tan. X. cot p} 

= 2cos. 2 h 1 = 2 cot 2 X. C ot 2 p 1 

.. tan- 3 X. cot. p = 2 cot 2 p tan. 2 X 
tan. 2 X 



Q 




tan. * X 



. . tan. 3 X + 




cot. p 



2 cot p = 0, 



from the solution of which cubic equation, tan- X is found. 

PROB. 18. Given the angle between 
two and three o clock in the horizontal 
dial equal to a. Find the longitude. 
From the triangle P R n, 
r . sin. P R = tan. R n . cot 30 

= tan. Rn. V3- 
From the triangle P R p, 

r . sin. P R = tan. R p . cot 45 

= tan R p. 
Voi. II. 




354 



PROBLEMS 



. . tan. n p = tan. a = tan. Up 11 n 
_ tan. R p tan. R n 
1 + tan. R p . tan. R n 



sin. X. (V 3 I) 



1 + 



sin. * X V 3 



sn. 



PROS. 19. In what longitude is the 
angle between the hour lines of twelve 
and one on the horizontal dial equal 
to twice the angle between the same 
hour lines of the vertical sun dial ? 

From the triangle P R n, 
sin. X = cot. 15 . tan. R n 

From the triangle p N m, 

sin. p M = cot. 15 . tan. N m 



= cos. X = cot- 15 tan. 
sin. X 



R n 



2 

tan. R n 




cos. X 



=r tan. X 



tan. R n 
~2 

Rn , Rn 

tan. -- + tan- 



1 tan. 



Rn 



1 tan- * 



Rn 



tan. 



Rn 



PROB. 20. G?n;e M<? altitude, latitude, and declination of the sun, Jind 
the time. 

cos. Z S cos. Z P . cos. P S 



cos 



;. h = 



sin. Z P . sin. P S 
sin. A sin. L . cos. p 

cos. L . sin. p 
cos. L. sin. p + sin 



or 



cos. L. sin. p 
_ sin, (p L)+sin. A 
cos. L . sin. p 

A -f P ~ 



sin. L. cos. p 

..- - .1 ..- 




A + L 



cos. L sin. p 



COS. 



h /cos. ( 
2 = V- 



FOR VOLUME III. 355 

the form adapted to the Lo- 



garithmic computation, or, see Prob. ( 1 8). 

PROB. 21. Given a star s right ascen 
sion and declination. Find the latitude 
and longitude of the star. 
Given 

y b, b S, L. S b 7 right angle 

.. find L. S 7 b and S 7. 

.-. /L S 7 a = S 7 b Obi. 

.*. S 7 is known, ^ S 7 a is known 

and S a 7 is a right angle, 
/. find S a = latitude 

7 a = longitude. 

Given the sun s right ascension and 
declination. Find the obliquity of the 
ecliptic. 

P S being known P 7 = 90, . S P 7 
= R A, 

. . in the ASP 7, .87? is known. 

.-. obliquity = 90 S 7 P is 
known. 

PROB. 22. In what latitude does the 
twilight last all night ? Declination 
given. 

(Twilight begins when the sun is 18 
below the horizon in his ascent, and 
ends when he is there in his descent, 
lasting in each case as long as he is in 
travelling 18.) 
R Q = H E = colat. = b Q + b R 

= D + 18. 
.-. 90 18 D = L 

= 72 1 D. 
(See Prob. 16.) 






356 PROBLEMS 

Find the general equation for the hour at which the twilight begins. 

Z 



E 




Let the sides P Z, P S, Z S, be a b c. 

(a + b + c 
o ^ 
2 

inensin.* =r __.-. 

H 



a J sin. f - 



or 



sin. a. sin. b 
/colat. + p -f 108 

. I - ! 

2 
sin, cotan. + p + 108 



sm.^ r . _. colat. J 



Sm 2 = 



2 



p) 



II 



cos. L . sin. p 

PROB. 24. Given the difference be 
tween the times of rising of the stars, 
and their declinations: required the lati- -, 
tude of the place. 

Given P m, P n, and the A m P n 
included. 

From Napier s first and second ana 
logies, the z. P m n is known, 
. . P m C = complement of P m n is 
known, 

.-. P C = 90, P m is given, and the 
/. P m C is found, 
/. P R =r latitude is known. 

PROB. 25. Given the sun in the equa 
tor, also latitude and altitude: find the 
time. 

Given 
Z P, Z S, P S = 90 find the A Z P S. 




FOR VOLUME III. 



357 




PROB. 26. The sun s declination = 8 
south, required the latitude, when he 
rises in the south-east point of the 
horizon, and also the time of rising. 
P S = 90 + 8, Z S = 90, L. S Z P 
= 45 + 90. 

Find Z P, and the A Z P S. 

PROB. 27. Determine a point in E Q, 
that the sum of the arcs drawn from it 
to two given places on the earth s sur 
face shall be minimum. 

Let A, B, be the spectator s situations, 
whereof the latitude and longitude are 
known. 

Let E Q be the equator, p the point 
required ; a b = difference of the lon 
gitudes is known. Let a p = x. 
.-. p b = a x. Let L, L be the la 
titudes. 

In A A a p, r . cos. A p = 
cos. L . cos. x. 

In A B b p, r. cos. B p = 
cos. L . cos. a x, 
.-. cos. L . cos. x + cos. L . cos. (a x) 
= max. 

.-. cos. L . ( sin. x) . d x + cos. L . X 
,sin. (a x). ( d x) = 0, 
.-. cos. L . sin. x = cos. L . sin. a. cos. x cos. L . cos. a. sin. x. 

Let sin. x = y 

.-. cos. L . y = cos. L . sin. a. V 1 y 2 cos. L . cos. a. y 
.. transposing and squaring 

cos. 2 L. y 2 2. cos. L. cos. L . cos. * y 2 + cos. * L . cos. 2 a. y * 

= cos. * L . sin. 2 a cos. 8 L . sin. * a y *, 
.*. y* = &c. = n. and y = V n. 

PROB. 28. To a spectator situated within the tropics, the sun s azi 
muth will admit of a maximum twice every day, from the time of his leav 
ing the solstice till his declination equal the latitude of the place. Re 
quired proof. 

a b the parallel of declination passing through Capricorn. 

Z3 




358 



PROBLEMS 



From Z a circle may be drawn touch 
ing the parallel of the declination till 
this parallel coincides with Z. .-. every 
day till that time the sun will have a 
maximum azimuth twice a day, and at 
that time he will have it only once at Z. 

(Also the sun will have the same azi 
muth twice a day, i. e. he will be twice 
at f.) 

PROD. 29. The true zenith distance 
of the polar star when it first passes the 
meridian is equal to m, and at the se 
cond passage is equal to n. Required 
the latitude. 

Given b Z = m, a Z = n, 

Z P = colat. = . m + n. 

PROB. 30. If the sun s declination 
E e, is greater than E Z, draw the cir 
cle Z m touching the parallel of the de 
clination, 
/. R m is the greatest azimuth that day 

If Z v be a straight line drawn per 
pendicular to the horizon, the shadow 
of this line being always opposite the 
sun, will, in the morning as the sun 
rises from f, recede from the south point 
H, till the sun reaches his greatest azi 
muth, and then will approach H; also 
twice in the day the shadow will be upon 
every particular point, because the sun 
has the same azimuth twice a day, in 
this situation. .. shadow will go back 
wards upon the horizon. 

But if we consider P p a straight line or the earth s axis produced, the 
sun will revolve about it, /. the shadow will not go backwards, 
r. cot. Z P q = tan. P q. cot. P Z, 




or 



cot. (time of the greatest azimuth) = tan. p. tan. L. 

All the bodies in our system are elevated by refraction 33 , and depress 
ed by parallax. 



o 



FOR VOLUME III. 359 

.-. at their rise they will be distant from Z, 90 + 33 horizontal pa 
rallax. 

A fix d star has no parallax, /. distance from Z = 90 -f- 33 - 

PIIOB. 31. Given two altitudes and 
the time between them, and the decli 
nation. Find the latitude of the place. 

Given Z c, Z d, P c, P d, L. c P d. 

From A c P d, find c d, and L. P d c. 

From A Z c d, find L. Z d c, 

.-. Z d p = c d P c d Z, 
.-. From A Z P d, find Z P = colat. 

PROB. 32. To find the time in which 
the sun passes the meridian or the hori 
zontal wire of a telescope. 

Let m n equal the diameter of the sun 
equal d" in space. 

V v : m n : : r : cosine declination, 
m n 




Q 



.-. V v = 



radius 1, 



cosine declination 
= d". second declination in se 
conds of space, 

/. 15" in space : 1" in time 

d" second dec. 




: : d" second dec. : 



15 



=r time in seconds of passing the merid 

Hence the sun s diameter in R A = V v = d". second declination. 

(n x = d" = sun s diameter) 

V v : m n : : r : sin. P n 

m n : n x : : r : sin. x n P 

V r : n x : : r 2 : sin. P n . sin. Z n P, 
r 2 . n x r 2 n x 



.-. V v = 



sin. P n. sin. Z n P 

r*. d" 

cos. X. sin. azimuth 



in. ZP. sin. P Zn 



sin 



r . d 



1 U 

. . time of describing V v = -rr-. * : : r- 

15 . cos. X. sin. azimuth 

which also gives the time of the sun s rising above the horizon. 

Z4 



360 



PROBLEMS. 




PROS. 33. Flamstead s m.elhod of determining the right ascension of a 
star. 

LEMMA. The right ascension of stars 
passing the meridian at different times, 
differs as the difference of the times of 
their passing. 

For the angle a P b measures the dif 
ference of the times of passing, which is 
measured byab = ay by. 

Hence, as the interval of the times 
of the succeeding passages of any fixed 
star : 360 (the difference of its right 
ascensions between those times) : : the 

interval between the passages of any two fixed stars : to the difference of 
their right ascensions. 

Let A G c be the equator, ABC 
the ecliptic, S the place of a star, S m 
a secondary to the equator. Let the sun 
be near the equinox at P, when on the 
meridian. 

Take C T = P A, .-. the sun s de 
clination at T = that at P. Draw P L, 
T Z, perpendicular to A G c. 
.-. Z L parallel to A C. 
Observe the meridian altitude of the 
sun at P, and the time of the passage 
of his center over the meridian. 

Observe what time the star passes over the meridian, thence find the 
apparent difference of their right ascensions. 

When the sun approaches T, observe his meridian altitude on one day, 
when he is close to T, and the next day when he has passed through T, 
so that at t it may be greater, and at e less than the meridian altitude at 
P. Draw t b, and e s, perpendiculars. 

Observe on the two days before mentioned, the differences b m, s m, of 
the sun s right ascension, and that of the star. 
Draw s v parallel to A C. 

Considering the variation of the right ascension and declination to be uni 
form for a short time, v b (change of the meridian altitudes in one day) : o b 
difference of the declinations) ::sb (=sm bm):Zb. Whence Z b. 
Add or substract Z b to or from T m. Whence Z m. Add, or take the 




FOR VOLUME III. 



361 



difference of, (according to circumstances), Z m, L m, whence Z L, 

I OQ ?7 T 

.. gives A L, the sun s right ascension at the time of the first 

/ 

observation. 

.-. A L + L m = the star s right ascension. Whence the right ascen 
sion of all the stars. 

PROB. 34. Given the altitudes of two known stars. Find x. 
Right ascensions being known, .. a b 
=: the difference of right ascensions, is 
known, 

.-. L a P b is known. 
.-. From AsPff, s P is known, 

and a s, 

From AZsu, z.s<rZis known, 
.-. L Z a P is known, 
from A Z a P, Z P is known. 



O 




Q 



PROB. 35. Given the apparent diameter of a planet, at the nearest and 
most distant points of the earth s orbit. Required the radius of the planet s 
orbit. 




D oc T. ; D greatest, jy nearest diameter. 

distance 

.-. D : D : : E P : E P 

::EP E C : C P 4 E C, 
.-. D C P + D E C = D C P D E C, 
D + D 



.-. C P = E C 



D D 



362 



PROBLEMS. 



PROS. 36. Given the sun s greatest apparent diameter, and least, as 101 
and 100. Find the excentricity of the earth s orbit. 



rad" 



the sun at 



the earth s orbit - 



100 : 101 : : S P : S P : : C P C S : C P + C S 
.-. 100 C P + 100 C S = 101 C P 101 C S 
.-. 201 C S = C P 
C P 



.-. C S = 



201 



, on the same scale of notation. 



O 




H 



PROS. 37. Two places are on the same meridian. 

Find the hour on a given day, when 
the sun will have the same altitude at 
each place. 

Z Z , two zeniths of places, .-. Z 2! is 
known, S the place of the sun in the 
parallel a b, Z S = S Z . 

From S draw perpendicular S D, 
.-. Z D = Z D, 

Z Z 

/. P Z + -g- = P D, is known, 

P S is known, z. S D P right L, 
. . L D P S = hour is known. 

PROS. 38. Find the time in which 
the sun passes the vertical wire of a te 
lescope. 

Meridian = the vertical wire, 
.*. the time of passing the meridian = 
the time of passing the vertical wire. 

Take m n = the sun s diameter = d. 

V v : m n : r : cos. declination, 

V d r 

cos. dec. 

.. V v converted into the time at the 
rate of 15 for 1 = the time required. 

PROB. 40. If a man be in the arctic circle, the longest day = 24 hours, 
the shortest = 0. 




FOR VOLUME III. 



363 



P Z = obliquity = Q R, 
.-. Z R = P Q = 90 
Z H = P Q = 90 

.. H R is the horizon, and the 
nearest parallel touches at R, 

.-. the day = 24 hours, and the far 
thest parallel touches at H, 

.. the day = hours. 

PROB. 41. Given the sun s meridian 
altitude = 62, midnight depression 
= 22. Find the longitude and declina 
tion. 

Qa = bQ 

or Ha H Q = R Q Rb 
= H Q R b, 
Ha+ Rb 



2 



= H Q = cos. x 



= 42, .-. X = 48, 

.-. D = 62 42 = 20. 

PROB. 42.Given the sun s declination, 
apparent diameter, altitude, and longi 
tude. Find the time of passing the 
horizontal wire of a telescope. 

s = the place of the sun. 

Take s n in a vertical circle = the 
sun s diameter = d. 

Draw n a parallel to the horizon, 
V v : a s : 
as : us : 
.-. V v : d : 

.-. V v = 

sin. cos. X sin. azimuth, 

verted into the time, gives the time re 
quired 



: r : cos. dec. 




: r : sin. n s P, 




: r 2 : sin. P s sin. n s 


P 


: r 2 : sin. Z P sin. P 


Z s, 


d r 2 


rn 



B 





364 



PROBLEMS 



PROB. 43. Given the longitude, 
right ascension, and declination of two 
stars; find the time when both are 
on the same azimuthal circle, and also 
of the azimuth. 

Given P S, P S , and L S P S = 
difference of right ascension. 
. z, P S S is known, 
L. P S Z is known, 
and Z P given, and P S given, 
.. . L. P Z S, is known = azimuth, 
and Z P S = time for the first star, 
or (Z P S + S P S ) = time for the 
second star. 

PROB. 44. Given the longitude and 
declination. Find the time when the 
sun ascends perpendicular to H R. 

D must be greater than X, or a Q 
greater than Z Q. 

Draw the vertical circle tangent to 
the parallel of declination, at d. 

P Z given, P d given, Z. P d Z is a 
right ., 

.. L Z P d is known. 




O 




PROB. 45. Find the length of the 
longest day in longitude = 45. 

Q d = obliquity, 
.-. P d = 90 obliquity = P c, 

Z P = 45, 

Z c = 90, 
. 2 hours is known- 




Q 



FOR VOLUME III 



PROB. 46- Find the right ascension 
and declination of a star, when in a 
line with two known stars, and also in 
another line with two other known 
stars. 

The star is in the same line with S, S , 
and in the same line with s, 0, 
.-. in the intersection s 



O 




PROB. 47. The least error in the time due to the given error in altitude 
= b". Find the longitude, 
n x is the given error in altitude, 
V v : m n : : r : cos. declination, 
m n : n x : : r : sin. x n P. 

V v : n x : : r 2 : sin. P n sin. Z n P, 

, 7 n x r 2 

V v 

sin. P n sin. Z n P 

n x r 2 

sin. Z P sin. P Z n 
- n x r 2 

cos. X sin. azimuth 
.. V v is least when the sin. azimuth 
is greatest, or the azimuth = 90, i. e. the prime vertical 
n x r 2 




.. b == 



.-. cos. X r= 



cos. X 

n x r 3 



PROB. 48. Given two altitudes, and 
two azimuths of the sun. Find the longi 
tude. 

Z S is known, Z S is known, L. S Z S 
= difference of the azimuth, 

.*. L. Z S S is known, 

.-. L Z S P = Z S S 90 is known, 

.-. Z S P, Z S, S Z P, known, 
find Z P. 




366 



PROBLEMS 




PROB. 49. Near the solstice, the declination a longitude, nearly. 

r sin. D = sin. L sin. 7, 
.-. r d (D) cos. D = sin. 7 d (L) cos. L 

d (D) __ sin, y cos. L 
r d (L) : cos. D 

sin. 7 cos. 90 d (L) 

= S * . since D 

cos. 7 

may be considered the measure of 7, 
= tan. 7 sin. d (L) 
= tan. 7 d (L), since d (L) small, 

d(D) tan. 7 
i ,-rL = = constant quantity, 

- d (D) ad (L) nearly. 

PROB. 50. Given the apparent time T of the revolution of a spot on 
the sun s surface, find the real time. 

Considering the spot as the inferior planet in inferior conjunction, 

T = p where P equals the earth s periodic time, p equals the planet s, 

.-. T P T p = P p, 
TP 

PROB. 51. The sun s declination equal 8 south, find the latitude of the 
place where he rises in the south east point, and also the time of his 
rising. 




Z c = 90, P c = 98, L. c Z S = 135, 
whence Z P, and L. \\ 



FOR VOLUME III. 



367 



O 




PROB. 52. How high must a man be raised to see the sun at mid 
night ? 

Z P = R Q. Take P d = Q b 

.-. b d = 90>. 

Draw x d to the tangent at d, 
/. if the person be raised to Z x, he will 
see the sun at b, 

A d C b = 90 = x C R, 
.. x C d ~ R C b measured by R b 
given. 
.-. in the rectilinear A x d C, L. x. d C 

= right angle, 

L. x C d being known from the dec. 
C d = radius of the earth. 

.. C x being known, 
.. C x 90", or Z x is known. 

PROB. 53. Given the latitudes and 
longitudes of two places, find the straight 
line which joins them. They lie in the 
same declination of the circle. 

V v : A B : : 1 : cosine declination, 

.. A B is known, 
and the straight line joining A, B, is the 

A B 

chord of A, B, = 2 sin. - . 

i| 

PROB. 54. A clock being properly adjusted to keep the sidereal time, 
required to find when y is on the meridian. 

P 






Observe the sun s center on the meridian, when the declination = x y, 
is known, 



368 



PROBLEMS FOR VOL. III. 



- x y 7 = right angle 
x 7 y = I, being known, 
x y is known. 

Whence y y = time from noon to 7 being on the meridian, or from 7 
being on the meridian to noon, whence two values of 7 y are found. 
If the declination north and before solstice the > value gives the time, 

after < 

If the declination south and before ]2+<l 

after 1 2 + > 



PROB. 55. Given the sun s declina 
tion and longitude, find his right ascen 
sion, his oblique ascension, his azimuth 
and amplitudes and the time of his rising, 
and the length of the day. 
7 C is given, from A c C d, c d is given ; 
I. and right angle, find c d. 
.*. C 7 = R A, C d = oblique asc n . 
and C d measures z. C P c, 

. . 90 + C P c = time of rising, 
2 (90 + C P c) = length of the day. 



(Thelwall.) 




369 



NOTES. 



To show that (see p. 16-) 

x d y v cl x cl x 

- J - 



2./C4X X 2A*- 

d t 



cl t 

* 1 

Not considering the common factor -T , we have 



/ -<ix)i 

J 



2 . /t X 2 . ,<i (x d y y d x) 

= ((* + l* + tt" + . . .) [p (x d y y d x) 

+ y! ( X d y y d x ) + A*" (x" dy" y" d x") + &c.} 
= A*t (x dy y d x) + ^ ( X d y y d x ) 

+ / /2 (x"dy" y"dx") + &c. 
+ /V (x d y y d x + x d y _ y d x ) 

+ p. ft" (x d y y d x + x" d y" y d x") + & c . 
+ vf p" (x d y y d x ) + ( x " d y" y" cl x") 

+ t>! t* " (x d y y d x + x" d y " y " d x ") + &c 
+ [S ft" (x x/ d y" y" d x".+ x /x/ d y" y " d x 7 ") + &C. 

&c. 

the law of whicli is evident 
Again, 

2 . A*y X 2 . /(* d x 2 . , x X 2 . ^ d y 
= (^ y + A y + / y" + ....) (t* d x + // d x + ^ d x" + .... & c .) 

(,<* x + /V x + ^" x" + ....) (IL d y + ^ d y + p" d y" + ..... ) 
= //.* (x d y y d x) /V (x d y y d x ) &c. 
+ i* (* (y d x x d y + y d x x d y) 

+ ^/// (ydx" xdy" + y" d x x "d y) +Sc c. 
+ ^ ^ (y d x" x d y" + y" d x x" d y ) 

-f ft t jJ" (y d x " x ; d y " + y " d x x " d y ) + &c. 
v + &c 2, 

VOL. II. 



370 NOTES. 

Hence by adding together these results the aggregate is 
p p. (x d y y d x + x d y y d x + y d x x d y + y d x x d y) 

+ /A p" (x d y y d x + &c.) + &c. 
ft ft" (x dy y dx + x" dy" y" dx" + y dx" x d y" 

+ y" d x x" d y ) + &c. 
&c. 
But 

xdy y dx + x dy y dx + ydx xdy + y dx x dy 
= dy (x x ) + d x (y y) + d y (x x) + d x (y y ) 
= (x - x) d y d y) (y y) (d x d x) ; 

and in like manner the coefficients of /A /// , ft ft " // ft", p! ft ", 

&c. are found to be respectively 

(x x) (d y" d y) (y" y) (d x" d x), 
(x " x) (d y" d y) (y " y) (d x " d x), 

( X " _ x ) d y" d y ) (y" y ) (d x" d x ), 
(x" x ) (d y " d y ) (y " y ) (d x " d x ) 
&c. 

Hence then the sum of all the terms in ft ft , PI*" /* ft", (* I*" 

n" ft ", ft ft"" is briefly expressed by 

2 . ft of f(x x) (d y d y) (y y) (d x d x)J 

and the suppressed coefficient ^-- being restored, the only difficulty of p. 
16 will be fully explained. 

That 2 . ( -r- ^ = 0. &c. has been shown. 
\d x/ 

2. To show that /( 2 2 . (* d x X 2 . P d * x) = (2 . t* d x) * 

page 17. 

3. ^ d 2 x = /^d 2 x + /* d*x + &c. 

= d . /* d x + d . ^ d x + &c. 
a? d (^ d x + fi! d x + &c.) 
= d . 2 . , d x. 

Hence 

/(22.^dx X s./t*d 8 x) =/2.2A6dx X d.s.,dx 

= (2 . ^ d x ; 2 
being of the form/ 2 n d u = u *. 



NOTES. 87 1 

3. To show that (page 17). 

2 . ,<* X 2 . ft (d x * + d y 4- d z 2 ) 

{(s.^dx) 2 + (2.,ady) + (2.A*dz) J 
= 2.^ $(dx dx) 2 + (dy dy) 2 + (dz --dz)j. 

Since the quantities are similarly involved, for brevity, let us find the 
value of 2 . p X 2 . i* d x 2 (2 . i* d x) 2 . 

It = (ft + ft + A*" + ) G* d x 1 + At d x 2 + /// dx //2 + ....; 
((L d x + A* d x + A*" d x" + ....) 2 ; 

Consequently when the expression is developed, the terms ^ e dx e , 
ft 8 dx 2 , /"* d x" z , &c. will be destroyed, and the remaining ones will 
be 

^ / (d x 2 + d x * 2 d x d x ) = A* X (d x d x) * 
" /^"(dx 2 -f dx //8 ~ Sdxdx") =t*(*"(dx." dx) 2 

^ / (d x ! + d x" 2 2 d x d x /x ) = // p" (d x" d x ) 9 
AtV w (d x /2 + d x*" 2 2 d x d x ") = (jJ n " (d x" d x ) 2 

At" A* /7/ (d x" * + d x " 2 2 d x" d x ") = ^ p!" (d x d x ;/ ) * 
&c. 

Hence, of the partial expression 

2 . p X 2 . /- d x 2 (2 . /* d x) ! = 2 . ft- (! (d x d x) 2 . 

In like manner 

2 . A* X 2 . Ai d y ? (2 . A* d y) = 2 . AV A* (d y d y ) * 
2 . A* X 2 . A* d Z 2 (2 . A* d z) 2 = 2 . A* A* (d z d z) 2 

and the aggregate of these three, whose first members amount to the pro 
posed form, is 

2 . A*/ Ud x d x) 2 + (d y d y) 2 + (d z d z) 2 ] 



4. To show that (p. 19.) 

/& x 

2 "p = 3x x 

2 . H (f V ) 3 

nearly. 

It is shown already in page 19 that 

3 \ 2 



372 NOTE S. 

x x 3 x r x v 

But since x / = x x\ y/ = y y\ z / = z z\ by substitution 
and multiplying both members by ,., we get 

.. v .. ,, o v \ q v \ 

i-^- O- ; A. * . y *OJw 

nearly. 
Similarly 

nearly. 

&c. 
Hence 

// V ^J II. Y ^i V^ ^ v^ 

[A x* a r* A /\ i\ \ \i 

^ o~ - ~* r\ r* ~~~"" ~~, \~\ ^ I -^ 2 /^ X "T" V 2 JBI T ~r Z 2 /^ *2 / *T" "/ \~\ IP 2 !!** 

s (s) d ) (s ) 

But by the property of the center of gravity, 

2 . /* x = 0, 2 . ^ y = 0, 2 . ,</, z = 0. 
Hence 

a, X 3 X v 



5. To show that (p. 22.) 

- c! 2 x +^d z y + Z cPz = ds gel 

S r 

and that 

x /d Qx y /d Q\ z /d Qx /d Q 

H ~ 



First, we have 

xd 2 x + yd y + zd 2 z 

= d (x d x + y d y + z d z) (d x 2 + d y 2 + d z *). 
But 

x ! + y s + z 2 = r, 

xdx + ydy + z d z:= ftl^ 
and because 

X = | COS. X COS. V 

y = o cos. ^ X sin. v 
z = sin. d. 



NOTES. 373 

. . d x 8 + dy * = [<1 (P cos. (?) . cos. v cos. 6 X d v sin. \\ * 

+ {( cos. d) sin. v + g cos. d v cos. v} 8 
= (d . g cos. d) 2 + g * d v 2 cos. * t>, 

.-. dx + dy + dz s = (d . g sin. 0) 2 + fd.gcos. 4) 2 + j 2 d v 2 cos, 2 * 
= dg i + 2 dd* + g*dv 2 cos. 2 0. 

Hence, since also 

d . d P = d 2 + g d 4 , 

x d 2 x + v d 2 y + z d 2 z , 2 . . 2 

- L_ - ! - a 2 g d v * cos. * 6 g d . 


Secondly, since j is evidently independent of the angles 6 and v, the 

three equations (1), give us 

/d X N x 

(-=-)= cos. 6 cos. v = , 
\ d g / f 

/d y\ y 

( , - ) = cos. ^ sin. v = ^ , 
\d o / f 

/d z\ . z 

( - - ) = sin. 6 = 

Vfl g/ g 

Hence 



x /d Q>. y /d Q\ L 

s vdir; " \_ijy " 



cl z 



^\ . ( l >| r y\ 4. r ^ f c z \ 
y viyj lay UT; va p fc 

But since Q is a function of (observe the equations 1), and g is a fuiic- 
tion of x, y, z, viz. Vx. 2 -f- y 2 + z s , 



But 



\ 

x / 



x 

and like transformations may be effected in the other two terms. Conse 
quently we have 

. n , /d x\ /d Q\ /d y\ /d Q 

Q : > (d7) (Sr) + d (dl) (~d~y 

Hence and from what was before proved, we get 



374 NOTE S. 



dt 






6. To show that xd 2 y yd 2 x = d (f z dv cos. 8 0), and that 



First, since 

x d 2 y = d.xdy dxdy 
y d 2 x = d . y d x d x cl y, 
.-. x d 2 y y d 2 x = d . (x d y y d x). 

But from equations (1), p. 22, 

d y = sin. v . d (g cos. 6) + g cos. . cos. v d v 
. ! x sin. v . d (* cos. 0) g cos. tf sin. v d v, 

/.xdy = sin. v cos. v . d (g ^ S " tf) + g 2 cos. 2 tf cos. 2 v d v 
y d K = sin. v cos. v . ^ -- g * cos. 2 ^ sin. 2 v d v J 

the difference of which is 

g 2 cos. 2 J X d v. 
Consequently 



l 



cos. 



Secondly by equations (1) p. 22, we have 

d v\ 

- ) = p cos. t) cos. v =: x 
d v/ 

d x\ . . 

-j ] =r P cos. <? sm. v = y, 
d v/ 

d Qx /dQx : . /clrx /d Qx /d_xx /d_Q 



/ x /jx : . /cjrx / x /_xx /_x 

<x Uy/ "~ y vix; U vJ VTryy vi vJ U x r 

y 

But since dividing the two first of the equations (1) p. 22, we have ?- 
= tan. v, v is a function of x, y only. Consequently, as in the note pre 
ceding this it may be shown that 



NOTES. 



= &) (a?) 



Hence 



IT" 

7. To find the value of (-7-7 )in terms of , v, 6, (see the last line but 

(1 9 / 

two of p. 22) 

Since d is a function of x, y, z, we have 

d Qv /d Q\ /dxx /d Qv /d JN /d Q\ /d 





But from equations (1) p. 22, we get 



(dtf) = - e sin- <J sin. 
d x 



.= f COS. 0. 

Hence multiplying the values of 

/d Q\ /d Qv 

CJT) J VdyJ 

d 2 x d 2 y d g z o 

by the partial differences we get 

Q\ 1 

z g cos. d - y . i sin. a sin. v d 2 x sin. 6 cos. v 



d t 2 
Now the first term gives 

g cos. D . d " z = ? [d" y sin. 4) cos. d + 2 d P d d cos. 2 

+ o cos. 2 dd 5 t? ^dd 2 sin. d cos. t)| , 
and the two other terms gives when added, by means of the equations (1) 

_ sin. 6 2 ain. 6 

cos. d " cos. ...* * " ^ 



NOTES. 
But 

d(ydy + xdx) = d.(d.x*+ y) = $d(cos., 
= d [g cos. 6 d (g cos. 0)} 
= (d . o cos. 0) 2 + f cos. 6 d * (e cos. 0) 
and 

d x s + d y 2 = (d . g cos. 6) * + p cos. * 4 . d v 8 . 
Hence 



cos. 



sin. 6 c 
~ ~ coll * cos< 6 J2 ^ cos *)"~ f s cos.tf dv s ] 

= g sin. 6 [d z ( cos. 0) cos. d d v *] 

= g sin. 6 [d* g cos. 2 d s d S sin. d d z 4 sin. 4 

d 6* + d v 2 g cos."*}. 
Adding this value to the preceding one of the first term, we have 



d X led** + 2&S&* + S&v* sin. cos. 6} 

.r.dl 8 d v . , 2 f d ? d^ 

- *d^ + * ff dT* sin cos ^ + dt* 

the value required. 

8. To develope , in terms of the cosines of 6 and of itsmul- 

-i ~f~ e cos. 

tiples, see p. 25. 

If c be the member whose hyperbolic logarithm is unity, we know that 

c v- 1 . c - g v- 1 
cos. , = + 



which value of cos. 6 being substituted in the proposed expression, we 
have 



1 2 c * <~ 



e cos. 6 ec 2tfv - 1 + 2c flv ^- 
2 c e ~ 



X 



~_ ^_ 2 

C 8< -1+-C 



But since 

^ f) 



NOTE S. 377 

gives 



C B v-i = . 1 +.. / JL _ i - 7 1 -- ^1 e2 V 
e V e 1 Ve+ e / 

and since, if we make 

1 (1 - Vl-e^ = X which also = j^-^-, 

we also have 

1 



_(l + VI -e*) = JL; 



the expression proposed becomes 

1 2 c <=T~ 

v 



1 + e cos. e 



2X 
= X 



e (1 + Xc*^- 1 ) (I + Xc- 

2X / 1 _ X c - 

= e (1 X 2 ) X U + xc V^T~ j + x - 



But 

x 
e 

and 



1 + VI e 2 



1 + e cos. d V(l e 2 ) 

which when = v w is the same expression as that in page 25. 
Again by division 



and 



Taking the latter from the former, we get 

T^ = 1 ~" X(c Vf ~ 1 + C ~ ( V ~ 1 ) + ^(c 2 ^ 1 +c-2" 
= 1 2 X cos. d + 2 >. cos. 2 2 X 3 cos. 3 & + &c. 



378 NOTE S. 

and substituting for the value v *, we get the expression in page 

9. To demonstrate the Theorem of page 28. 

Let us take the case of three variables x, y, z. Thja our system ol 
differential equations is 



in which F, G, H, are symmetrical functions of x, y, z ; that is such as 
would not be altered by substituting x for y, and y for x ; and so on for 
the other variables taken in pairs ; for instance, functions of this kind 

Vx* + y 2 + z* +~tY(x y 4 xz+xt+yz+yt + zt), 

q 

(x y z + x z t + y z t), 
log. (x y z t) and so on. 

Multiply the first of the equations by the arbitrary , the second by /3, 
and the third by y and add them together; the result is 

= II (x + V + 7 z) + G (a ~ + 8 1? 4- ~ ? 



4. F (a ^ ~ X 4- 8 i -~^- 
d t * d t 2 

Now since a, 8, y, are arbitrary, we may assume 

which gives 

d x d y d z 

d x djr d 8 / 



d 2 x d 2 x 
and substituting for x,-j-^-, y^- , their values hence derived in the first 

of the proposed equations, we have 



NOTES. 379 



_A x y - X = 0. 

a a 

In the same it will appear that 

ax + /3y+7Z = 

verifies each of the other two equations. It is therefore the integral of 
each of them, and may be put under the form 

z a x + b y 

in valuing only two arbitrages a and b, which are sufficient, two arbitra- 
ries only being required to complete the integral of an equation of the 
second order. 

In the equations (0) p. 27. 






= H, G = and F = 1 

and f 3 being = (x 2 + y * -f z 2 ) 1 is symmetrical with regard to x, y, z. 
Hence the theorem here applies and gives for the integral of any of the 

equations 

z = a x + b y, 
see page 28. 

Again, let us now take four variables x, y, z, u ; then the theorem pro 
poses the integration of 



. 
= II x + G + 



d t d t 



Multiplying these by the arbitraries a, j3, 7, S and adding them we get, 
as before 

= H (a x -j- /3 y + 7 z + 3 u) 

" X L P. " 7 L , l l^ _L A _ 



, ^ / d s x d* y . cl ! z d u 

+ F a -~ + /3 - + 7 + a r 



380 NOTES. 

Assume 

ax + /3y + yz-f6u = 0. 

and upon trial it will be found as before, that this equation satisfies each 
of the four proposed equations, or it is their integrals supposing them 
to subsist simultaneously. As before, however, there are more arbitraries 
than are necessary for the integral of each, two only being required. 

Hence the interal of each will be of the form 



This form might have been obtained at once, by adding the two last of 
the proposed equations multiplied by y and d to the two first of them, and 
assuming the coefficient of H = 0, as before. 

In the same manner if we have (n) differential equations of the i-th order, 
the order involving the n variables x (!) , x (2 > . .. . x w , and of the general 
form 

d x w ^ d 2 x W d 1 x w d x (>) 

fa} i /"^ f T^ ** ^ i A*-* A ,VlA. x/ 



O 
= 






we shall find by multiplying i of them (for instance the i wherein first 
s = 1, 2 . . . . i) by the arbitraries a ( l \ a.(\ ..... a W; adding these results 
together and their aggregate to the sum of the other equations ; and as 
suming the coefficient of H = 0, that 

B 0) x d) 4. a (2) X P> + .... a CO x W + X i + 1 + X s + 2 + ..... X a = 

\\illsatisfyeachofthe proposed differential equations subsisting simulta 
neously ; and since it has an arbitrary for every integration, it must be 
the complete integral of any one of them. 

This result is the same in substance as that enunciated in the theorem 
of p. 28, t inasmuch as it is obtained by adding together the equations 
whose first members are x W, x ( % &c. and making such arrangements as 
are permitted by a change of the arbitraries. In short if we had multi 
plied the i last equations instead of the i first by the arbitraries, and 
added the results to the n i first equations, our assumption would have 
been 



which is derived at once by adding together the n i equations in pige 
28. 



NOTES. 381 

If we wish to obtain these n i equations from the equatt. n (a), it 
may be effected by making assumptions of the required form, provided by 

so doing we do not destroy the arbitrary nature a V\ a P\ a ( ;) . The 

necessary assumptions do, however, evidently still leave them arbitrary, 
Those assumptions are therefore legitimate, and will give the forms of 
Laplace. 



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