, <-.s (2) 174 MR. ROBERT RAWSON ON SINGULAR where R,, R,...R, are functions of «x, y, which must satisfy the following partial differential equations, viz. aR, _, aR,» dia ay | a, _ | aR, de" ap | oh (3) ama oa im dy J) And c,, ¢,.... ¢, are arbitrary independent constants, which, however, may be made to satisfy (m—1)-fold relations. 3. An interesting case of the above is where each of the values of c,, ¢, .... €, 18 represented by the arbitrary constant (c); then (2) may be written (e+ R,) (c+ R,) .... (C+ R,)=0;. . 2 ee where R,, R,....R, are the roots of the primitive (4) with respect to the arbitrary constant (c). Equation (1) in its present form is not generally an exact differential equation of the primitive (4); the multi- plier, however, which will make it exact is readily obtained by differentiating (4) with respect to w, and will be as follows :— I (R,=8,)?...(B,—B,)*x (8. 8,)... Re - (5) x (R,—R,)*. .. (R,—R,)*...x (R,—R,)*... (R,—R,,)” Now it is proved by writers on the subject of singular solutions of differential equations that a singular solution SOLUTIONS OF DIFFERENTIAL EQUATIONS. 175 makes the multiplier, or the integrating factor, infinite; hence from (5) the singular solutions of (1) are fh)... (Be —R,) x (R,—B.)*.2. (RB, R,)* ee oh (Re yee. 2S. (6) Of course each of the factors must be equated to zero, therefore the theorem (A) is obvious. From (3), if R,=R, then will r,=7, &c., therefore the truth of theorem B is manifest. 4. In consequence of the importance of the principle of the condition of equal roots in the determination of sin- gular solutions of differential equations, another proof of it may not be deemed unnecessary. In the primitive (4) the functions Be eo 2. kb, Cal be made to assume various forms; put, therefore, E.=¢,4 70, BSo— Vw, 0-00 R,=v,+ VW; R,=v,— VW, BF Tos Be (8) R,=0,+ Vw,, Re=v,— Vw, . . ~ (9) &e., Sie. where V,, W,; V2, W2}3 V,, W,, &c. are functions of 2, y. The primitive (4) becomes by substitution {(e+v,)*—w,}{(e+0,)*—w,}{(e+v,)*—w,}...&e.=0, . (10) where (c) in each quadratic factor has a two-fold value. Differentiate (10) with respect to 2, eliminate (c), and reject the common factors; then —dv, dw, ; dv, dw, ay NG ae ree ae cal dx — ae | dx oe ge a “a dy a dy dyJ 176 MR. ROBERT RAWSON ON SINGULAR which is a differential equation having for its complete primitive equation (10). Equation (11) is not, however, an exact differential, as the multipliers which would make it so are ight ak gee Tae Vw, Vw, The first, second, &c. factors of (11) are evidently satis- fied by the relations &e. W, = O, (UP cos OF a Se; pay fae; which are, therefore, singular solutions of (11) if they are not contained in the primitive (10) by giving a constant value to (c). By this assumption of the values of R,, R,... &c., it is seen at once that the differences R,—R,, R,—R,.. . &e., are made to vanish by the conditional equations (12) ; hence the truth of theorem (A). Since the equations (12) make the roots, with respect to (p) mm the quartic factors which compose (11) equal, therefore the theorem (B) is also proved. The singular solutions here considered are of the enve- lope species, as they may be obtained by eliminating (c) between (10) the complete primitive and MY —o, It will I I be seen that the integrating factors ——, —=—, &c. are Vu, Vw, made infinite by singular solutions, which is a well-known property. 5. In this paragraph there will be proved some of the properties of singular solutions by means of the condition of equal roots. | , SOLUTIONS OF DIFFERENTIAL EQUATIONS. 177 Differentiate (10) with respect to (c), reject the common factors; then dy +2Vw, dy +27, de dv, dw de —dvy, dw / I ae I 2 wh 2 +2 ire a £2 VWF di eee Ore Fs oie (hd) Now the condition of equal roots gives w,=0, w,=0, &c., therefore = in (13) is evidently zero. It may be shown ! te Av: P 1n a similar manner that — is zero also. dc In the case of the envelope species of singular solutions dy _ dx : “although not necessarily equivalent, do not lead to conflicting results” (Boole’s Diff. Eqs. p. 146, 2nd ed.). For a particular hypothesis with respect to the form of =o are equivalent. Boole states that such, the complete primitive, viz. (O(e—X)\"de—y—o0, . . . s. (a4) where Q is a function of x, c, which neither vanishes nor becomes infinite when c=X, Boole has proved that for a singular solution (Z) =infinity. Boole also states “ that inquiries which are scarcely of a sufficiently elementary character to find a place in this work indicate (with very high probability) that this character is universal and inde- pendent of any particular hypothesis, and that it consti- tutes a criterion for distinguishing solutions of the envelope species from others” (Boole’s Diff. Eqs. p. 163, 2ud ed.). The proposition, viz. 7 =infinity, which seems to have originated with Laplace, and was subsequently investigated by Lagrange, Cauchy, Poisson (see Boole’s Diff. Eqs. pp. 172,174, 2nd ed.), may be proved generally as follows :— 178 MR. ROBERT RAWSON ON SINGULAR Differentiate each factor in equation (11), and reject the common factors; there results for the first factor by drop- Ping the affixes, dw oN — dv a d*w aig. dy y dx! /w ” dedy dady pay == +2 ee dy dw — I —~d’vy dw +2 OO ay ae tay dy) Jat? Yay ae +2 Ne ad i= dy dy For a singular solution w=o, then (15) becomes pete se ay PD sc eg (16) (Pee) 2 ~\dedy dy dx dw is a finite quantity, and v a function of zw, y. When, es dp _o : however, the above quantity is zero, then ae an inde- terminate quantity, and by Boole’s Diff. Eqs. p. 24, 2nd If ed., v will be a function of w, and thereby it is probable that the solution will be a particular integral. 6. As the differential equation of the first order and of the nth degree is composed entirely of the simple factors of a differential equation of the first order and degree, it will be necessary to consider the latter form only. Let dy Fr a ° ° e ° rc e e (17) be a differential equation where 7 is a function of 2, y. ¢ y fr SOLUTIONS OF DIFFERENTIAL EQUATIONS, 179 Put Ce Ore (18) for the complete primitive of (17), where (c) is an arbi- trary constant and (R) a function of x, y, which, however, must satisfy the partial differential equation dk dR (19) og oe 7. The function (R) may be taken so as to satisfy the equation R= oar). a 4 Pe] where v, Z, w are functions of x, y. Substitute this value in (17) and (18); they become respectively fois I dy oa | a a ane aoe eee =O, (2 1) i +l fiw) * CAD Zp) Oy al mia! Pe ae (a Of course fw) 2 = Sf). It is clear, therefore, that (21) is satisfied by the relation Oy oe ee ie ae © (23) where h is any arbitrary constant if f"(h) =infinity. The differential equation (21) 1s, therefore, satisfied both by the complete primitive (22) and by the equation (23), which is a singular solution of (21), if it is not contained in (22). by giving a constant valuc to (c). - This form of the complete primitive, to which all primi- tives may be reduced, gives the characteristic singularity to the differential equation (21), which is derived from it 180 MR. ROBERT RAWSON ON SINGULAR by ordinary differentiation, and the term singular solution of (21) can be justified only on the ground that it serves admirably to distinguish the two kinds of solution referred to in (22) and (23). In the general theorems when a solution is said to be singular, it is meant to be so only when it is not contained in the complete primitive by giving a constant value to (c) independent of #2. (See Boole’s Diff. Eqs. p. 163, 2nd ed.) 8. The following examples will illustrate the general formule :-— Let | ( dy\* a ay TL Saiae aaa? Vary +27xe+y=1. . (24) Required the singular solution, if any. This equation admits of the form LL hak ORS rer \(¢ ) 3 = 2Ve+yt+I fie glee (25) The condition of equal roots is evidently Va +y =i. oh ee eg ee” (26) But this is not a solution of (24), therefore the principle of equal roots fails to give the singular solution. Again, dp I dy” Va+y = infinity, if e+y=0, And this is the singular solution required. The complete primitive of (24) is (c—2)*— (y+ Va+y) (c—a) +yVaty=0. . (27) The condition of equal roots is ee SOLUTIONS OF DIFFERENTIAL EQUATIONS. 181 which is not a solution of (24), therefore the principle of equal roots applied to the complete primitive fails also to give the singular solution. The elimination of (c) from (27) by the condition dy dx : : Se and an leads to the equation (28). The usual methods, therefore, of finding the singular solution from the complete primitive fail in this example. The primitive (27) admits of the form (c+ Va+y—2) (e+y—2)=0, from which we have the two equations c+ VLX+y=2, c+y=2. The first represents a series of parabolic curves, and the second a series of straight lines to which the singular solution «+y=o is perpendicular. Let dy _y y” dx ae ee (29) oe “sin— Required the singular solution if there is one. This example is given by Sir James Cockle, F.R.S. &c. (See ‘ Quarterly Journal of Mathematics,’ No. 54, 1876.) Its complete primitive is e-+0— cos” =o. ee eee"! (20) Compare this equation with (22), then v=a2, z= —1, and S(w) =cos Therefore f'(w)=—sin ; which does not become infinite for any value of z or y. The equation z=0 satisfies (29), but it is a particular 182 MR. ROBERT RAWSON ON SINGULAR solution, as may be inferred from (30) by expanding x E cos - and taking c=1. a a ace ae d (t)= The condition «=o gives Tamed and dea =0. Boole says, “ If Zz =infinity leads to a solution which does not involve (y) in its expression, nothing is to be inferred whether it is singular or not. Then the proper Pest is c =infinity.” Why not? the inference here is that z=o is a particular solution of (29), which has been already proved. The equation y=o does not satisfy (29), as may be seen from the following development :— Since xc x sin — UG iy See i ce i a 6y e show ee Le. Ti. Sia aig? POT 360 aM isracyt te ; Be infinity. x SIN co | a ek ee Therefore Ign ntinity in (29) when yo. g. The following particular cases of (21) and (22) are interesting. Let Jiwy=log wu, Ss. fw) =<. SOLUTIONS OF DIFFERENTIAL EQUATIONS. 183 Equations (21) and (22) become by substitution dv dz, ; dy f aoe a og WwW we - dx au ‘dz. ma w+ eit 7 aes, Esa dy * dy °® * ly GU NOG WO a ~*~ (32) Now (31) is satisfied by w=o, which, however, is not a singular, but a particular solution by taking (c) equal to infinity. From (32) Wie wire SH erat ee were w=e * = co ee? *, w=0, when c=infinity. Let I f(w) =w”, where (”) is an integer. +. fl(w) = pie e Substitute these values in (21) and (22), then re a), Bete dy de de (os (33) da Oe ee en a ee “ Cry ewe = Otis ead s,s ..(34) This equation, viz. (33), is satisfied by w=o, which is a singular solution, if it is not contained in the complete primitive (34) by giving a constant value to (c). 184 MR. ROBERT RAWSON ON SINGULAR . Again, let J(w)=cos w, then f'(w) = —sin w; substitute these values in (21) and (22), then (9 + & cos 7 a ae dy Co ee sinw ~ dx ~~ ee w) I iG dw? - . (35) dy dy a sin w “dy c+0+zc00sw=0. . |... ; Now w=o is not a solution of (35), since sin w is zero, and no value which can be given to (w) will satisfy the equation sin w=infinity. Let dy _ylogy 0 gp 2 lt nn v _ logy v =O. +. .. is its complete primitive. This example is from Boole’s Diff. Eqs. Supp. vol. p. 17. In equation (34) put v=o, z=—logy, n=—1, and w=x; then =o is a solution which satisfies (37), but which is not contained in (38) by giving a constant value to (c), therefore it is a singular solution of (37). Boole rejects the solution w=o because it does contain in its expression (y). For what reason this rejection is made does not appear. By reference to equations (31) and (32), put v=o, — =<, and w=y, then y=1 1s a solution of (37), but it is a particular solution by giving c=o in the primitive y=e", Boole has obtained the solution y=0 in two places, viz. ——— ~ SOLUTIONS OF DIFFERENTIAL EQUATIONS. 185 Diff. Eqs. p. 161, 2nd ed., and supp. vol. p. 17, by means of the condition - =infinity, and thereby plunged him- self into a complete mathematical labyrinth, from which his great genius would not enable him to extricate him- self. He states that ‘the value of (c) is not wholly independent of (z),” and “regards y=o as a singular solution ;” whereas y=o is a particular solution, where c=—infinity. For the value of (c) to be dependent, in any degree, upon the value of (x) is in direct opposition to the spirit of the definition of a singular solution. By paragraph (9) it appears that no differential equa- tion, whose primitive is of the form c+v+zlogw=o, can have a singular solution. . 10. There are three questions which demand special consideration in the subject of singular solutions of differ- ential equations, viz.— 1. The singular solution as derived from the complete primitive. 2. The singular solution as derived from the differ- ential equation. 3. Given a solution of a differential equation, is it a singular or a particular integral ? These questions have to some extent been replied to in paragraphs 1 to 5. 11. With respect to the first question, it may be observed that every complete primitive can or cannot be put into the form (22). When the former is possible then the sin- gular solution is obvious from the primitive itself, but if the latter prevails then the conclusion is that there is no singular solution. It may be noticed here that other terms of the form SER. III, VOL. VIII. fo) 186 MR. ROBERT RAWSON ON SINGULAR 2,f,(w,) may be added’ to the primitive (22), thereby giving rise to other singular solutions, w,=hA, &c. - 12. The second problem is more difficult to answer than the first, and seems to have been first generally considered by Laplace himself. (See Boole’s Diff. Eqs. p- 174, 2nd ed.) Let WV +n =0 ae be the differential equation whose singular solution is required, (r) being a function of 2, y. By reference to equation (21) f’(w), being the special object of inquiry, is the factor which makes (21) an exact differential equation, therefore (39) admits of the form dy | fi(w)r _ ae =Q. +)». San If, therefore, (40) is an exact differential equation, then £ p(w Ja 7 A (w)r}, or dp _ f"(w) ret (4.1) dy fi(w) \' dy da 4 Since = dp. ar p=-—r and iy = iy Now by paragraph (7) the singular solution w—h=o makes f'(h) =infinity, therefore f'(w) must, for a singular solution, necessarily be of the form fwy= 2, whereof the function ¢(w) does not vanish for w=A, SOLUTIONS OF DIFFERENTIAL EQUATIONS. 187 Differentiate (42) with respect to (w), then Mp) — PY) _ 9), ah ese substitute the values of f'(w) and f"(w) in (41), then ap~ pie) tia) ay ge) 8 When, however, w=h, which is the case for a singular solution, then the equation (43) becomes infinite on the ae Camere tee oo ( dw “ dexter side; hence ie providing sce ae does not vanish. The condition, therefore, which satisfies 7, =infinity, and also satisfies the differential equation (39), may be a singular solution, and will be if it is not contained in the primitive by giving a constant value to (c). It is not difficult to show that the complete primitive of (39), in terms of its integrating factor f'(w), is tf (u)dy rp wae (L™ aydamo. . (43) 13. The solution of the third question is more difficult still than either the first or second. It will be seen by what follows that Boole’s solution to this question is not regarded as altogether satisfactory (see supp. vol. pp. 28, 2020): The assertion by Boole that F(#, 0) is a constant, or rather F(x, 0) 1s not a function of x, for a particular solu- tion is not universally correct. The theorem, therefore, in problem (10), p. 28, supp. vol., is faulty in proof; it is also difficult in its application and uncertain in its utter- ance. Observe the following example :— 02 188 MR. ROBERT RAWSON ON SINGULAR Let (zt) - (ety ts) Sh —a0(e w—y—1)=0, - (44) whose complete primitive is ct ofa +a*—y—log (w—y)} + («*—y) {a—log (wy) }=0. « (45) Now z—y=0 is a solution of (44); is it a singular or a particular solution? The primitive can be put in the form {e—a+log (r—y)}{ce+y—x2*}=0. . . (46) Therefore c—2x-+log («—y) =o, from which ae € a gee ol ae} when c=infinity. Hence x—y=o is a particular solution, the constant (c) being infinite. It remains to eliminate (y) from (44) and (45) by means of WL HY wae er Then n (=) +(20—w— 1) — (291) = 0, - (48) c*—c{a* +w—log w} + (#*—a#2+w)(a—logw)=o. . (49) Now w=o is a particular solution, and satisfies (48), but the primitive (49) still remains a function of x, which is in opposition to the principle used by Boole (supp. vol. p- 28). 14. Boole’s proof of a theorem from Euler. First eliminate (y) from the given differential equation, and its complete primitive by the relation w=¢(a, y) ; then dw ade =P(v,w), » « « + » (50) SOLUTIONS OF DIFFERENTIAL EQUATIONS. 189 whose complete primitive is c+F (a, 05) =O} ° ° e e e (51) C” dw a) (ae, w)- OFS COMP ah) yo in'< (52) makes w=0 a singular or a particular solution. Differentiate (51) with respect to x, then d dw dx B(x, w) de. a Hence from which there results aw d Seal ee eens (7), . s (53) r ar ( , w) d \ : } da § 30) The integral on the dexter side, as given by Boole, is equal to H. F(z, w), “where (H) is some value inter- mediate between the greatest and least values of I upon £¥(a, w) within the limits of integration.” See supp. vol. p. 29. This integral, however, is regarded as unsatisfactory, therefore the following is proposed in its place, viz. d ( i we (es Bw) Se (54) ‘p(x, w) AVC, Bw) 190 MR. ROBERT RAWSON ON SINGULAR where (8) is a proper fraction (see Todhunter’s Diff. Cal- culus, p. 81, third ed.). Nothing as yet has been said with respect to the conditional equation w=¢(«, y) by means of which (50) and (51) have been obtained. The above general integral obtains whether w=o is a solution of the differential equation or otherwise. Between the limits zero and (w) the integral becomes (; dw a5 ne) ' go OD (2, 2) £ F(a, 0) From paragraph (13) it appears that F(z,0) may be and frequently is a function of 2 when w=0 is a particular solution. Boole’s argument in supp. vol. p. 28, is not universally correct. The inferences from (55) are as follows :— When w=o and F(z, o) 1s a function of a, then wo Gree sae. aye) But whether w=o is a singular or a particular solution the equation (56) does not determine. When w=o and F(z, 0) is a constant, then wd Vy 5-3 oe This equation, which is indeterminate, gives w=o as a particular solution if £ FE, o) is finite. Boole’s ex- ample, supp. vol. p. 31, confirms the truth of (57) as log oe d, is indeterminate when y=o. There is nothing whatever to show, in Boole’s demonstration, that w=0 is a singular solution of the envelope species. SOLUTIONS OF DIFFERENTIAL EQUATIONS. 191 15. To find, without the aid of the complete primitive, whether a solution w=o of a differential equation of the first order and of the nth degree is singular or particular is a problem the answer to which is not by any means obvious, and, if it be solved, which is doubtful, by the process given by Boole (see supp. vol. p. 30), the solution is accomplished by a troublesome transformation, accom- panied by a definite integral, which, by the bye, is often impracticable. The following answer to this difficult problem may be interesting. Reduce, by algebraic methods, the given differential equation into its quartic factors, of which the following is one, viz. (Si+r)(B+s)=0, area ak aa Sieh) where 7, s are given functions of 2, y. If the solution w=o be of the envelope species the results of the substitution of (w) in (r) and (s) will be equal. Tt will now be necessary to compare (58) with the first factor in equation (11), paragraph 4, omitting the affixes ; then —dv dw —dv dw Vine — Fe ar(2vu—S2),. . . (59) Vite — 7 =o(2V0s — 7); taker.’.(OC) 192 MR. ROBERT RAWSON ON SINGULAR From these equations (v), the only unknown element in the primitive c+v+~Ww=o, when Vw is given, can be easily found by the usual process, as given by Boole (see Diff. Eqs. pp. 48, 49, 2nd ed.). If, however, the solution w=o has unfortunately dropped a factor, it will be a serious matter, in many cases, to pick it up, as wz*=o0 must be used in (61) and (62) instead of w=o, where z is an unknown quantity to be determined. It is therefore obvious from the primitive that the solu- tion w=0 cannot be particular if (v) contains x or y, except in the case of (v) being a function of (w). If the dexter side of (61) and (62) are functions of a, y, the solution w=o is singular and of the envelope species. Example :— xp’ —2yp+42=0 (from Glaisher), . . (63) I woe = pe + —/y* —42*=0. ~~ @& Compare this example with (58), (61), (62); then r+s= 2Y is S pny 4 a” eas x’ =4, @ y 4 z The condition of equal roots of (63) is y= +2a, which is a solution of (63); then w=y —4a@, dw re aioe tw, dy y- SOLUTIONS OF DIFFERENTIAL EQUATIONS. 193 Hence w=0o is a singular solution of (63) of the envelope species. 16. By reference to equation (11) and by dropping the affixes to the variables (v) and (w) it is easy to obtain Oo” fdwry ) ( dvdy dwdw (4e(G) A iy) je +27 | 4G iy de dy 3 +40(2) - = == (0 Mar oe (64) This equation, which is of the second degree in (p), is expressed in terms of its primitive c+v+/w=o, and always has a singular solution w=o when (v) is not a constant. A differential equation of the second degree and the first order, which is not of the form (64), cannot have a - singular solution. Professor Cayley says (see ‘ Messenger of Mathematics,’ vol. vi. p. 23) that Lip*--2Mp+N=0+ 4 = .. +. (65) has not in general any singular solution when L, M, N are rational and integral functions of a, y. If I apprehend correctly the meaning of this statement, it seems difficult to reconcile it with equation (64), the terms of which can always be made rational and integral functions of x, y by proper assumptions of (v) and (w). Now, equation (65) admits of the form Litp*+2MLp+M*—N=o... . . (66) The condition of equal roots is N=o, and this equation is a solution of (66) if aN dN ee (67) 194 MR. ROBERT RAWSON ON SINGULAR Substitute this value in (66) and it becomes LG pth +VN=o. . 1 This equation, which is of the second degree in (p), is expressed in terms of the differential equation (66), and has in general a singular solution N=o of the envelope species. Equation (68) includes all the particular examples given by Professor Cayley and Mr. Glaisher in the papers already referred to. 17. Professor Cayley has considered the particular case Lp*—-N=0. ... . «2 The condition of equal roots of (69) is N=o, which is a solution of (69) if N is a function of y only. The equation L=o is a solution of (69) if L is a func- tion of x only. Hence, examples 3, 4 given by Professor Cayley can have no singular solutions. Their complete primitives are respectively | c+29+(sin“y+y W1—y”)=0, . . (70) e+sin yty V1—y"*+ (sin @7+a2V71—2*)=0, (71) which are the only solutions of these two examples. Compare (69) with (57), then 7+s=0 and ‘i dv NdVw yp hiel L dy” e ° ° ° ° (72) adv L dVw dy = N dx’ e e ° ° e (73) SOLUTIONS OF DIFFERENTIAL EQUATIONS. 195 The conditional equation is af, [Navy _ af, [Lave ak an av eee? fc) Equations (72), (73) determine the nature of the solu- tions N=o and L=o. This equation must be satisfied by a proper value of Ww, or otherwise equations (72) and (73) cannot be integrated. Example :— p =i1—77 Grom Cayley). vs 1/40 G75) The condition of equal roots is y=+1, which is a solution of (75), but whether it is singular or particular depends upon (72) and (73). In this case st+r=o, L=1, s=ViI-y’. The equation (74) is satisfied by Musi vtay. ©. << .- « (76) Then (72) and (73) become | ee dp OY SD 2, ave COS & ey ee from which V=Y COS 2, therefore the complete primitive is e+y cosxv+sinaV1—y*=0, and y= ae I is a singular solution of the envelope species. Example :— p (i—z*)=1—y* (from Cayley). . «. (77) - The condition of equal roots is in this case v= +1 and 196 MR. ROBERT RAWSON ON SINuJLAR y= +1, which satisfy (77), but equations (72) and (73) will affirm the nature of these solutions. Put L=1—2’, N=1-—y’; then the conditional equation (74) is satisfied by Vw=¥V 1—2*.1—y’, and (72), (73) become From these two equations dere 4: Ee therefore the complete primitive is ctaytV1—a2.1—y*=0. . . . (78) Hence y=+1 and v=+1 are singular solutions of the envelope species. Equation (78) admits of the form c+ 2¢cay+y’+a°—I=0, ... 1 em from which y= —crt+Vi—cViI—2. . . . (80) It is seen from (80) that neither (c) nor (x) can exceed unity, for if they do (y) becomes impossible. Because (c*—1) is negative the curves (79) are a system of ellipses (see Todhunter’s ‘Conics,’ third ed. p. 239), whose principal axes and centre are the diagonals, and their intersection, of the square y= +1, y= +1. The conditional equation (74) is satisfied also by Vw=y Vi—# and Ww=2 Vi—y’, giving the primi- tives ee sa Bet be Ca - (81) e+yMi—a'+aVi-y=o. SOLUTIONS OF DIFFERENTIAL EQUATIONS. 197 Mr. Glaisher states the condition of equal roots of (77) to be 1—a.1—y*=0. All I can say at present is that I—z*=0 satisfies the condition of equal roots in a peculiar way. It may be stated that (77) can be replaced by its equivalent a—y) (3) =a". Boi Netrbeed rae (82) The roots of which, with respect to (=), are equal when = +1, a condition which satisfies (82). This principle can be applied to equation (65), with a view to show the cases in whick L=o may produce a singular solution as well as N=o. 18. Professor Cayley states that if Lp*+2Mp+N=o has a singular solution, it is either M*—LN=o or a factor of M*—LN equal zero. To this proposition may be added another, viz. : dM dM _ has a singular solution M=o, which does not appear to be included by the above proposition of Professor Cayley’s. 198 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT XVIII. On the Formula for the Intensity of Light trans- mitted through an Absorbing Medium as deduced from Lixperiment. By James Bortomury, B.A., D.Sc., F.C.S. Read October 17th, 1382. In a former communication an experimental method was _ suggested for testing the validity of an assumed law of intensity of light that has passed through an absorbing medium. The method was this: take two surfaces of different degrees of brightness, survey them through some absorbing medium, adjust the lengths of the columns so that the intensities shall be the same; then, if the law of absorption be true, the intensities will again be equal if both columns are increased by the same length. Some experiments which I made gave results in agreement with the theory. In these experiments surfaces of different degrees of whiteness were observed through a grey solu- tion. The error arising from the finite extent of the surface is small, and the mean intensity which we observe may be taken as the intensity of the central ray. But suppose we had started with no hypothesis as to the form of the function expressing the intensity of trans- mitted light, but had found, as an experimental result, that when the intensities are equal, they remain equal, when the columns receive equal increments; what form for the function might be deduced from such a result ? Suppose there are two lights of initial brightness, I, and I', respectively, then if # and y be the lengths of the TRANSMITTED THROUGH AN ABSORBING MEDIUM. 199 absorbing columns for the transmitted light in one case, we shall have ul So a ee 9 and in the other case i lO gee Wee See ea, (2) g@ being the unknown function which it is required to determine. Since these two intensities are equal, Dee Cea Mar oT" 2S auie Ti ee SK If both columns receive an equal increment, the inten- sities will again be equal. Let this common increment be denoted by «, then Vib (re) SVG YK). oe ee (4) These equations will still hold if y=o. In this case the duller surface is observed directly, and the length of the absorbing column over the brighter surface is increased until they are brought to the same intensity. Since when y=o, I',=I',, therefore ¢(0)=1. Equations (3) and (4) will become ME rae ee Nee re Ph eet ey, cia at OS) Le i) Pe ted Se 2 AMS) Expand (6) in terms of x, 1.4 $ (2) + a eae! ay i: =I (1 +4'(0 e+ £0) (0) x + &c.). Since the first terms of the expansion on each side are equal, the equation may be written 14 e+ S49 Ss PF JS (di (o)e+ a: = ke.) 200 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT Dividing both sides by « and diminishing « without limit, the resulting equation will be Lone =T'.6"(0) ; eliminating I, and I’, by (5), The integral of this is log (x) =$'(o)7+e, or d(x) =Ce® ©, When xr=0, dh (a) ='T, Therefore Cpe he Also as # increases, the intensity diminishes, therefore g'(0) must be some negative constant; let it be denoted by —m. ‘Then the equation becomes and equation (1) may be written 1S ieee Hence experiment leads to the same form for the function as the hypothetical form with which we started. If, in the above investigation, we had made the length of the column invariable, and x denoted a mass of some colouring-matter which undergoes no decomposition on dilution, we might have obtained experimentally the form of the function expressing the intensity of the light trans- mitted through a column of fluid of invariable length, containing a variable quantity of colouring-matter. TRANSMITTED THROUGH AN ABSORBING MEDIUM. 201 In the aboye remarks I have supposed we are dealing with homogeneous light, or with white light which has penetrated a medium containing soluble black in solution. To apply the formule generally we must prefix to them the sign of summation. In seeking @ priori the law of transmitted hght, we might have reasoned as follows, which involves less assumption than Herschel’s reasoning. Suppose we have a column of any length; conceive it divided anywhere into two lengths w and y by an imaginary plane. Jet I, be the initial intensity of light ; after penetrating the column 2 we shall have Ip diz): But if light of intensity I, penetrate a column of length y, the transmitted light will be I,¢(y), or by substitution I,d(z)¢(y). Since the length of the column is v+y, the emergent light will also be expressed by I,f6(v+y). Kquating these two expressions for the same quantity, there results | (2) $y) =$(w+y). It is well known that this functional equation is satisfied by an exponential form. SER. III. VOL. VIII. Ae 292 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER XIX. On the Intensity of Light that has been transmitted through an Absorbing Medium, in which the Density of the Colouring-matter is a Function of the Distance traversed. By James Bortomuey, B.A., D.Sc., F.C.8. Read October 17th, 1882. In previous papers it has been supposed that the absorbing matter was uniformly distributed throughout the medium. For such a case the laws I=Zae~™ and I=Sae “2 are applicable, ¢ denoting the length of the absorbing column, and g the mass of colouring-matter contained in it. In the present paper these laws are applied to the case of light passing through media of variable density. Such cases occur in nature; for instance, the atmosphere is of variable density, increasing as we approach the earth; also we might have coloured glass in which the colouring- matter is not uniformly distributed, or the case of a coloured soluble salt on which water is poured; the colour in the immediate vicinity of the salt is most intense, and gradually fades as the distance increases. The same reasoning will also apply very approximately to fluids containing in suspension very finely divided matter in layers of variable density, or to an atmosphere charged with very fine dust. For simplicity I shall suppose that we are dealing with homogeneous light, or with white light that has passed through a grey solution. Suppose that a ray has penetrated a length ¢ of a variable medium, and that the intensity is I when it falls TRANSMISSION THROUGH AN ABSORBING MEDIUM. 203 on a surface for which the coefficient of transmission is e-”, Let us take a plate of the medium of thickness Af, and let « “*4” be the coefficient of transmission at the upper surface. If I' be the transmitted heht, | Phe: Tee > Ie Expanding, and putting AI for I/—I, this becomes ATI a i es —mAt+m => Te, Ps > — (m+ Am) At+ (m+ Am) i — Xe. Ultimately we get COE SE)! oe re ae mr ae Now suppose m to be some function of ¢; if this be some integral form, then we may determine I. If light has passed through a homogeneous medium ¢ units long, then I[=I,e-™. If the length be unity, I=I,<-”. If the unit length contain g units of colouring- matter, we also have [+I,e~"2. Equating these two values of I, there results the equation ; Ne = wd. Let d be the density of the colouring-matter, then d will be proportional to g. If the unit of density be that due to the distribution of the unit mass through unit volume, then we may replace g by d. Now suppose the density variable and some function of ¢, so that we may write d=¢(t). Substituting in equation (1) we get leet = pr die a st (2) P2 204 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER Let x(é) =p (t)dt. Then, integrating equation (2), we obtain log I= — px (t) +c, which may also be written in the form Te: |, oo ae e (3) To determine the constant we must know the initial values of I and ¢. Suppose that when ¢=o0, [=I,; then T.=Ce™, and substituting for C in (3) we get i 1,¢°% Bes This will be the gencral expression for the intensity of light which has penetrated a length ¢ of a medium of variable density. I. Suppose the density to vary as the distance from the plane of incidence; then we have nt” yO Naa ae and {2 I=Ce “2, If I, be the intensity when ¢=o, we get TZ=C; This determines the constant. If for nt we substitute d, we obtain the following relationship between the intensity and the density at any point :— Bey | elise an, TRANSMISSION THROUGH AN ABSORBING MEDIUM. 205 II. As another example, suppose the absorbing medium to be an elastic fluid arranged in concentric layers about an attracting sphere of radius R, the law of attraction being that of the inverse square. Let ¢ be the distance of any point from the centre, then if f be the attractive force, m being a constant. It may be shown that the density at any point will be given by the equation Mm d= Ac, SH SEU A a ME cle aca (4) where a@ is a constant denoting the relationship of the density to the pressure, and A another constant which may be determined as follows :— Let D be the density at the surface of the sphere; then making f=R, we get A=De Hence we have m nr pO) =De et and for the intensity of the transmitted light we have m m AC eee ae TE (a) The value of C may be obtained by substituting for I its. value at the surface of the sphere, and after integrating the quantity under the integral sign, substituting R for ¢. - Equation (4) may be written in the form | eg ie d=De e827), 906 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER If r be small compared with R, we may write daDe ak Ga RB) Let the attracting body be the earth; then for m we may substitute gR*, and we obtain = T d=De*. J) 2 In this case the expression for the intensity becomes Ee 2 De i I= Ce To determine C make I=I, and r=o. Then “D Las > and the expression for the intensity becomes g late eek .@ This is the expression for the intensity of light which has passed through a length 7 of the atmosphere, supposed to be of uniform temperature, neglecting the differences in the force of gravity at different heights. If the law of decrease of temperature with altitude be given, then @ in (6) must be multiplied by some function of 7. When + becomes large the expression (7) tends towards a limiting value a ar fates" The atmosphere, especially in the lower layers, is never free from dust; of the whole atmospheric absorption a considerable portion would probably be due to this. TRANSMISSION THROUGH AN ABSORBING MEDIUM. 207 The relationship of the density at any point to the pressure is given by the equation i= her és 3 In the previous cases the ray has been supposed to pro- ceed through the atmosphere from the earth; if the path of the ray is through the atmosphere towards the earth, the formule will require a little alteration. Let # be the distance of the luminous object from the earth, and I, its brightness at that distance, @ its distance from any layer of the atmosphere below it, and + the distance of that layer from the earth. Then we have h=7 +06. Hence the expression for the density becomes —2(h-6) d=De * and the formula for the intensity of the transmitted light becomes Ae gee) j=! ay > ae If, in this formula, we make =A, and then make h large, we have for the limiting value of the intensity ee a I=I,e° 9. In equation (7) the light is supposed to proceed through the atmosphere vertically. When this is not the case the expression for the intensity becomes more complex. Ia treatises on geometrical optics may be found the following differential equation of the path of a ray of light which has passed through a medium, of which the refractive 908 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER index at any point is a function of the distance from a fixed point, dO C dr ~ WV wr —C aa From this equation | we obtain be cae soe Or; VS 7 y>— CP Let R be the earth’s radius, and z the apparent zenith distance at the place of observation ; then if the refringent power of a gas be constant, so that u* —1=x«d, we obtain ds _ rM (1 + «d) dr Vy* (I + xd) — R* sin?z. This leads to the following differential equation for deter- mining the intensity of the transmitted light at any point :— ge ge dlogI _ a iy +KDe “%e" dray tian a r(r+KDe © Tagan) — — R’ sin*z III. We might also have the absorbing matter distributed through the medium in such a manner that the same value for the density would recur. Suppose, for instance, that the colouring-matter is distributed in parallel layers, and that the law of density is given by the equation d=m—nsin ft, where ¢ denotes the distance from some plane, and m and n are constants, m being greater than n, the initial density will be m, and the value will recur whenever sin ¢ vanishes. The density will have a maximum value m+n whenever ¢ is of the form 2yT+3—, where v has any TRANSMISSION THROUGH AN ABSORBING MEDIUM. 209 positive integral value including o, also the density will have a minimum value m—n whenever ¢ is of the form ir oie . ° . uk > where v has any positive integral value including o. A plate of such a medium, cut off by two parallel planes perpendicular to the layers of colouring-matter, when viewed against the light, would present the appear- ance of light and dark bands gradually increasing and diminishing in intensity. In this case we have x (2) =|(m —n sin t)dt, =mt+ncosf, and the formula for the intensity becomes _— Cee ae cos Z) Wo dewimime CC; make 7—o and I=; then je Corte: Hence we have jf Le” eae oe cos i) At successive layers of maximum density we shall obtain T I ote Te” gee (aon+ 3 ay where for vy must be substituted o, 1, 2, &c. The above may be written more briefly fea Hence on emergence from successive layers of maximum density the values of the intensity are in geometrical progression, the first term being I,C and the ratio e-?, 210 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER At successive layers of minimum density we get rateneen (ort) which may be more briefly written b= 1 Cer where for v must be substituted in succession 0, I, 2, &e. Hence the values of the imtensity on emergence from layers of minimum density constitute another geometrical series, of which the first term is I,C’ and the common ratio e~?, The nature of the absorption may be more clearly understood by representing the variables by coordinates. Let ordinates represent intensities and abscisse the distances traversed by the hight. Then, if we start with light of intensity I, and allow it to traverse a medium for which the coefficient of transmission is constant and of value e~"”, the resulting curve will be a logarithmic curve, and, if through the same medium light travels of initial intensity I,¢*”, the curve will again be a logarithmic curve lying above the other; then the curve of intensity, after traversing a medium of variable density such as we are now considering, will be a sinuous curve always situated between the above-mentioned curves and touching them alternately, the poimts of contact with the upper curve corresponding to those values of the abscissze which make cos # vanish and the points of contact with the lower curve corresponding to those values of the abscissee which make cos @ equal to unity. IV. By means of the general expression for the inten- sity of transmitted light we may also solve inverse ques- tions such as What must be the law of density in order that the intensity may be some assigned function of the distance traversed? Suppose, for instance, we require the TRANSMISSION THROUGH AN ABSORBING MEDIUM. 21] law of density in order that the intensity may vary as some inverse power of the distance: in this case so that the general equation becomes m —=Ce {* : Differentiating, we obtain == Ce by, since ox) = $(t). But Hence we get the following equation :— n t = pd, so that d varies as the distance inversely. Since we get, by eliminating f¢, l=m (Ayan 3 n this gives the relation between the density and the inien- sity at any point. Note.—In several papers on colorimetry which I have read before this Society, I have frequently had occasion to refer to the hypothesis that as the length of an absorbing 212 INTENSITY OF LIGHT THROUGH AN ABSORBING MEDIUM. medium increased according to the terms of an arithmetical ratio, the intensity of the light diminishes according to the terms of a geometrical ratio. J had not then been able to trace this hypothesis back farther than the writings of Sir John Herschel, but had some grounds for supposing that it might have been given earlier, and more especially by Bouguer. Lately, after much inquiry, there has come into my hands a small treatise, entitled ‘ Essai d’Optique sur la Gradation de la Lumiére, par M. Bouguer, profes- seur royal en Hydrographie,’ Paris, 1729. From this work it appears that the honour of having first announced the hypothesis belongs to Bouguer. In many otherwise excellent treatises on Physics and Optics the subject of absorption of lght is either neglected or scantily treated, and the claims of Bouguer seem to have nearly passed out of recognition ; yet he may assuredly claim herein a position correlative with that assigned to Snell, or Huyghens, or Newton in those depart- meuts of optics of which they laid the foundations. The treatise contains no experimental verification of the hypo- thesis, nor any suggestions for carrying out such experi- ments. He was aware that the subject afforded a vast field for future inquiries, and with regard to his own work he modestly states, in the preface, “C’est vrai que mes recherches sont poussées si peu loin qu’elles laissent encore un vaste champ & tous ceux qui voudront perfectionner cette matiére. Mais ne scait-on pas que les Arts les plus simples ont eu leurs différens ages, et que ce seroit comme étoufer dans le berceau les découvertes qu’on peut faire dans la suite, que de mépriser toutes les premieres tenta- tives, sous prétexte que ce ne sont encore que de foibles commencemens ? ”” In the last section he has also considered the intensity of light which has passed through a medium which is not ON A NEW VARIETY OF HALLOYSITE FROM SERVIA. 213 of the same density throughout. By geometrical reason- ing he arrives at the conclusion that the curve of intensity (the gradulucique, as he terms it) has this property: its subtangent multiplied by the density is equal to a con- stant. Expressed in the language of the differential calculus, this gives rise to a differential equation similar to the one which I obtained by a different method, and gave last session in a paper read before the Society on the intensity of light which has traversed a medium wherein the density is some function of the distance traversed. Except in the consideration of the intensity of light which has passed through the atmosphere, Bouguer has made very little use of this highly general theorem ; for, says he, in most cases we do not know what is the law of density. This may be so, but by assuming the density to be some function of the distance, we may deduce some interesting and valuable results. XX. On a New Variety of Halloysite from Maidenpek, Servia. By H. E. Roscoz, LL.D., F.R.S., Presi- dent. Read January 22nd, 1884. _ THis mineral, one of the few peculiar to Servia, was given to me by Mr. James Taylor, lately a resident at Maidenpek. It is avery soft (h=2°5) whitish green non- crystalline mineral, having a conchoidal waxy fracture. It is translucent in thin films, but opaque in mass, and adherent to the tongue. Its specific gravity is 2°07. On exposure to air it loses a portion of its combined water, 214 ON A NEW VARIETY OF HALLOYSITE FROM SERVIA. and becomes of a dead white colour and more opaque. The greenish tint is due to the presence of small quantities of copper oxide (1°11 per cent.). The following analyses show that this is a more highly hydrated variety than most of the specimens of Halloysite hitherto examined, and that it corresponds to the formula AJl,0,2810, + 5H,0. Analysis of Halloysite from Maidenpek :— i. cy Mean AURO Sols sadeo tee st AEE Se ogee Weer 8 snes 32°69 DLO,» che see eame S7ESOM Sette Wi Mery ee 37°64 elk Geyer pe ees BE GOK ascik aus are. | meecetees 28°43 Ou@ (3i.. 0 tas TOUS. wees ekeees SS arenes I'l I00°10 = 99°87 pl 0 RRP eee tetas Rae Me er epiae a 32°86 TO eee eatelees os cen shale eitcca- ae eri Se 38°37 ER SO 5 eetiapeins. oeeos ae ace tne eee 28°77 100°00 A specimen of a similar mineral from the same locality was found by Tietze to contain :— EO (BS GJS) i cser aban aaen: Shree ccm soc 25°20 ROB rca teced Malone tranche un ew nieme nes 44°96 BIO) ote ce eaae pa sieteonenmer wears 29°50 99°66 Corresponding nearly with the formula Al,O,3S8i0, + 6H,0. Showing a distinct difference in the relation of alumina to silica from that existing in the specimen in question. ON THE INTRODUCTION OF COFFEE INTO ARABIA. 215 XXI. On the Introduction of Coffee into Arabia. By C. Scuortemner, F.R.S. Read February sth, 1884. In two papers, which I read on April 3rd and October 16th, before this Society, I mentioned that the custom of drinking coffee originated with the Abyssinians, who culti- vated the plant from time immemorial. In Arabia it was not introduced until the early part of the fifteenth century ; before this time the beverage made from the leaves of the kat was generally used, and is stiil in use. _A few weeks ago I received a letter from Professor W. T. Thiselton Dyer, F.R.S., in which he says: ‘ Possibly the enclosed extracts from an old book of the last century may interest you. “The point is that the introduction of the use of coffee from Persia, in the 15th century, seems to have led to the neglect of kat. “Your interesting observation as to the absence of caffeine in the latter, would perhaps show that the change from one to the other had a physiological significance.” This appears very plausible. I hore to be able to obtain a larger supply of kat, in order to find out its active principle. The extracts which Professor Dyer sent me are as follows :— 7 ‘A Historical Treatise of the Original of Coffee.’ London, 1732 (pp. 308-310). 7 216 MR. C. SCHORLEMMER ON THE “ Jem al Adin Abu Abdallah, Mohammed Bensaid, sur- named Al Dhabhani (because he was a native of Dhabhan, a small town of Arabia Feelix), being Mufti* of Aden, a famous town, and part of the same country, about the. middle of the oth age of the Hegirah, and of the 15th of our Lord, had occasion to make a voyage to Persia. During his stay there, he found some of his countrymen who took coffee +, which, at first, he took no great notice of; but at his return to Aden, his health being impaired, and calling to mind the coffee which he had seen taken in Persia, he took some in hopes it might do him good. Not only the Mufti’s health was restored by the use of it, but he soon became sensible of the other properties of coffee ; particularly that it dissipates heaviness in the head, exhi- larates the spirits, and hinders sleep without indisposing one.” = The Arabian author adds, that they found coffee so good that they entirely left off the use of another lquor which was in vogue at Aden, made of the leaves of a plant called Cat, which cannot be supposed to be the The, - because this writer Says nothing which might favour that opinion. Since this was written Mr. W. Elborne, of the Owens College, called my attention to a paper by Mr. James Vaughan, Civil and Port Surgeon at Aden, who states that some estimate may be formed of the strong predilection which the Arabs have still for kat, from the quantity used in Aden alone, which averages about 280 camel-loads annually. He adds that he is not aware that kat is used in Aden in any other way than for mastication; from what he has heard, however, he believes a decoction * An order of Priests amongst the Mahometans, which may be called their Bishops. Tt Coffee first in use at Aden, the capital city of Arabia Feelix. INTRODUCTION OF COFFEE INTO ARABIA. Q17 resembling tea is made from the leaf by the Arabs in the interior™. 7 Mr. Vaughan gives also some abstracts from De Lacy’s ‘Chrestomathie arabe,’ in which it is stated, on the autho- rity of some Arabian authors, that coffee was not introduced into Arabia by Mohammed Dhabhani, as it was generally stated, but by the learned and godly Ali Shadeli ibn Omar. In the days of Mohammed Dhabhani, kat, which previous to that time was used, had disappeared from Aden. “Then it was that the Sheik advised those who had become his— disciples to try the drink made from the boonn (coffee- berry), which was found to produce the same effect as the kat, including sleeplessness, and that it was attended with less expense and trouble. The use of coffee has been kept up from that time to the present.” As the custom of drinking coffee originated in Abyssinia, it appears more probable that it was introduced into Arabia from this country, and not from Persia. My friend Professor Theodores has informed me that the beverage made from the boonn is called kahwa. This word is derived from ikha, dislike or distaste, 7. e. for eating and sleeping. eh * Pharm. Journ. Trans. xii. p. 268 (1352-53). SER. III. VOL. VIII. Q 218 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON XXII. On the Equations and on some Properties of Prajected Solids. By James Borromtey, B.A., D.Sc., F.C.S. Read March 18th, 1884. On a former occasion I brought before the Physical and Mathematical Section of this Society a proposition in projection, in which it was shown how, by the composition of two projections, namely, of that of a line on a line, and of that of a plane area on a plane area perpendicular to the aforesaid line, we could derive from a solid three solids with axes perpendicular to three planes at right angles and of variable volume, the variation being subject to the condition that the sum of the three volumes is constant and equal to that of the primitive solid. I now propose to solve the following problem: Given the equation to the primitive solid to deduce that of a derived solid. Let the equation to the primitive solid referred to three rectangular axes be SOME PROPERTIES OF PROJECTED SOLIDS. 219 Let ABC be the primitive plane, which is fixed in the solid, and DE an axis perpendicular to this plane, and which may be called the primitive axis. Let P be a point situated on the intersection of the solid by a plane parallel to the primitive plane. Draw PF perpendicular to that plane. In deducing the equations to the derived solid we might have taken a system of axes in the primitive solid as follows: take DE as one axis, and two straight lines per- pendicular to this, lying in the plane ABC and passing through the point D, as the other two axes. Then, if the equation to the solid be given, referred to these movable axes, we may deduce by geometry the coordinates of any point on the derived solid referred to the axes fixed in space Oz, Oy, Oz. We may, however, refer both the derived and the primitive solid to the same system of fixed aXes. | : Draw PG perpendicular to the plane zy; on PG take a length LG, so that LG=PF cosy, _y being the inclination of the primitive axis to the axis of z. Then L will be a point on the derived solid. Also we have ; | Ei —2) cos. DPS. DPF is the angle between PF and PD. PF is parallel to the primitive axis, and its direction cosines will therefore be cos a, cos 8B, cosy. Let a, b,c be the coordinates of the point D; then the direction cosines of the line DP will be 7 | PESO Cy b 26 -BD? es PD a; BD. Therefore we have (~—a) cosa+ (y—b) cos B+ (z—c) cosy , PD ‘ cos DPF= Q2 220 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON also we have 2=KG, Y =HG, z=PG. Let &, 7, denote the coordinates of the corresponding point on the derived solid, thus ¢=KG, m=IG, €¢=LG=PD cos DPF cos y. Hence we obtain E=2, n=Y; (1) ¢=cos y{(2—a) cosa+(y—bd) cos B+ (z—c) cosy}. Hence the Ps to the derived solid will be __ (§—a) cosa+(n—6) cos B #(6 ? cos*y * cos + If z be given as an explicit function of xv and y, say z= (2, y); then the derived surface will be ¢=cos y{ (E—a) cosa+(n—b) cos B—c cosy} + cos*v¢(F, 7). Hence the equation of the derived solid will be of the same degree as the primitive solid. In the figure we have taken as primitive plane, a plane intersecting the solid, and its projection as falling on the plane x, y; im this case that portion of the solid lying below the primitive plane will, when projected, lie below the plane 2, y. We might have taken as the primitive plane the tangent plane at the lower extremity of the primitive axis; in this case the whole of the solid, when projected, would be above the plane 2, y. SOME PROPERTIES OF PROJECTED SOLIDS. 221 As an example of the use of the above given relation- ships between the coordinates of the primitive and the derived solid, suppose we find the relation between their volumes. Then using V, to denote the volume of the derived solid between limits o and %, we have Med dae a rate)? (2) by substitution this becomes V,=Cos WANE (#—a) cos «+(y—b) cos 8+ (z—c) cos ytdy dz, a result which may also be written in the form Tos “y{\(z—e4 | #2—a) cosa+(y —bd) EES COS or finally in the form z V,=cos*y dx dy dz. o— t= cosa+(y—b) cosB cos y If, in equation (1), we make =o, we obtain - (~7—a) cosa+(y—b) cos B+ (z—c) cosy=o. ; This, being an equation of the first degree in 2, y, z, will represent a plane; it is, in fact, the equation to the primi- tive plane. Hence the lower limit in the above integral reminds us that the integration is to extend from the primitive plane to points upwards; but if V be the volume of the primitive solid between these limits we shall have G2 V=\\\ az dy dz. . (z—a@) cosa-+(y—b) cos B os cosy Hence . . : ie cos*7V. 222 DR.J. BOTTOMLEY ON THE EQUATIONS AND ON In a similar manner we might show that V,=cos eV, V,=cos 67, and so establish the relation given in a previous paper, V.+ V,+Vz=V. In equation (2) suppose that the limits of the inte- gration are €, and %,, so that the volume of the solid is \\(&.—-&) dédn. Let z, and z, be the corresponding values of the points on the primitive surface. Now, in integra- ting with respect to z, we regard x and y as constant; hence €,=cos y {(w—a) cos a+ (y—b) cos B + (z,—c) cosy}, €, =cos y{ (x—a) cosa+ (y—8) cos B+ (z,—c) cosy}; by subtraction we obtain €,— $= (2.—2,) cos*y 5 hence, by substitution, we obtain MG. —&) df d= cos*y {\(2,—2,) dev dy. As a particular case, consider the projected solid derived from a sphere (w~a)*+ (y—b)* + (2—0)*=r", Take as primitive plane a plane passing through the centre of the sphere, and as primitive axis a line perpendicular to this passing through the centre of the sphere; then the equation of the projected solid will be n*{ (—a)* + (y—b)*} + {2—n(1(a@—a) +m(y—b))}*=ntr*, (3) 1 being written for cos a, m for cos 8, n for cos y. +e To simplify the equation to this surface remove the SOME PROPERTIES OF PROJECTED SOLIDS. 223 origin to the point a, 4, 0, and let the new axis of x make an angle @ with the old axis, so that tan ¢= + - then the equation becomes 2° —22n@ VI —n* + nx" + ny? =n*r*. Now transform to new axis in the plane of 2, z, the new axis of wv making, with the old, an angle @, so that 2n tan 20 = —— 3 b (5 then we get the equation xe ae 2nt*r* 2n*r* ae +£=1. (4) + I+n’*—V/1+2n*—3n* I+n*+ V71+2n7—3nt 7 Thus the surface is an ellipsoid; its volume may readily be shown to be sanrs, that is m*xvol. of sphere. If n= +1, that is if the primitive axis be parallel to the axis of z, the equation to the ellipsoid becomes ety +e", which is the equation to the primitive solid referred to coordinates passing through its centre. If, in equation (4), we make n=o, the sanetnalle axis of the ellipsoid parallel to the axis of x will assume the indeterminate form =. On examination it will. be found to be r*, but when n=o the projected solid vanishes ; hence, just as it vanishes the ellipsoid becomes a spheroid, of which one of the axes is indefinitely small. In a former communication I stated that a sphere when 224 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON projected would give a spheroid of which the semiaxes are R, R cos y, R cos y, whereas in (3) an ellipsoid has been obtained. It will be observed, however, that there is something arbitrary in the mode of projection. The sole condition to which the projected solid is subjected is Adz =cos*7V ; but this condition is not sufficient to determine the equa- tion to the projected solid. If L be the length of the primitive axis, then if we draw two parallel planes distant Leosy, these planes being parallel to the plane of a, y, any solids terminated by these planes and having equal sections made by any plane parallel to them, will fulfil the above condition. Hence we have some choice as to the manner of laying the successive slices of the projected solid on one another; we have just found an ellipsoid as a particular case, but if the elliptical sections parallel to the plane of 2, y had been piled on one another, so that all the centres lay in the same vertical line, we should - obtain a spheroid of which the semiaxes are R, R cosy, R cos ¥. The locus of the centres in (3) may be got as follows. Let a section be made by a plane parallel to plane of 2, y and distant € from that plane; write also X + X, for x, and Y+Y, fory. Let X, and Y, be so determined that the coefficients of X and Y vanish. This leads to the con- ' dition ; : (X,—a) (nt*+n7l*) +n*lm(Y,—b)—fnl =o, (Y,—)) (n*+n’m*) +n*lm(X,—a) —&am=o. Hence, X,, Y,, Z, being the coordinates of the centre of the section, we have SOME PROPERTIES OF PROJECTED SOLIDS. -- 235 From these equations we obtain wea? VS 8 OF, Mita ca ths ies The locus of the centres is therefore a straight line parallel to the primitive axis; we might have drawn any arbitrary curve, either plane or of double curvature, terminated by the bounding planes parallel to plane of x, y, and piled up the successive slices, so that their centres lay on the curve ; the solid so generated would also fulfil the condition of a projected solid. As another example, consider the projection of the elliptic paraboloid z=Ax*+By’. The equation to the derived solid is z=n{ (~—a)l+m(y—b) —cn} +n? (Ax + By’). The volume of the primitive solid included between the _eurved surface and the plane 2h h \\\ dx dy dz. Az2-+ By? _ The plane, when projected, will give an equation of the form ee will be z=n{I(@—a) +m(y—b) —nc} +n7h. ~The volume intercepted between this plane and the pro- jected solid will be . n4U(x—a) +m(y—b)—ne} +-n2h dx dy dz, n{i(2--a)+m(y—b)—ne} +n2(Ax2-+ By2) 226 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON which is equivalent to n2h h \\\ dx dy dz or to a\\ dx dy dz, n2(Az2+ By?) 3 Ax2+4 By2 that is n*xvol. of primitive solid between the corre- sponding limits. The primitive solid and its z derivative being written in the form z=$(2,y), z=n{I(4—a) +m(y—b) —cn} + n*d(a, y). If we multiply the first equation by n* and subtract from the second, we get the equation 2(n*—1)+nle+nmy—n(la+mb+nc) =o. Hence, if the primitive and its derivative have any points in common they satisfy the equation to a plane. This plane is at right angles to the primitive plane, for @ being the angle between the planes, we shall have y 6 nl nm ee Cos hay aes Fuge PY I n — ?+m*+n*—I arc ys (“= 0. The relation V,=cos*yV is only a particular case of a more general theorem; for let ¢(&, 7) be any arbitrary function of & and 7, and let I, denote the integral ¢ \\\ve n) dé dy dt ; SOME PROPERTIES OF PROJECTED SOLIDS. 227 then, by substitution, we may show, I denoting the integral h(x, y) dx dy dz, e—-F=—® cosa+(y—b)cosB cosy I,=cos’yI. that As another example of the use of the substituted coordinates, suppose that between any two points of the primitive solid any arbitrary curve be drawn lying on the surface. et s be the length of this curve, and ds an element extending from the point 2, y, z to the point z+dzxr,y+dy,z+dz. Let s,, s,, s, be the lengths of the curves on the three derived solids passing through the points corresponding to those through which the curve on the primitive solid passes; then, ds,, ds,, ds, being elements of these curves, we shall have ds,= d&* + dn’ + de” ; by substitution this becomes ds, = dz’ + dy* + cos*y (dx cos a + dy cos 8 + dz cos y)*, and in like manner may be obtained ds, = dx* + dz* + cos*B(dx cos a+ dy cos 8 + dz cosy)’, ds7,= dy* + dz’ + cos’a (dx cos a+ dy cos 8+ dz cos y)*. By addition we obtain. ds; + ds, + ds; = 2 (da* + dy” + dz*) + (dx cos a+ dy cos 8+ dz cosy)*. (5) Let ¢ be the angle between the primitive axis and the 228 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON tangent at any point to the fixed curve on the primitive solid, then we have dy da dz cos 6= qs COS * + qs Cos B+ 7s 0°88 also ds* = dx* + dy* + dz’ ; hence the right side of (5) may be written in the form ds*(2+cos*). Since the curve is fixed on the solid, if we suppose the solid to move in any way, we shall have \ 2+ cosh .ds a constant quantity; if this be denoted by C, then we shall have throughout the motion § Vds2 + dst + dst=C. The lengths of the curves on the derived solids will vary as the primitive solid changes its position, but will vary in such a manner as to satisfy the above equation. By reference to the expression for ds,, it will be seen that it does not vanish when cosy vanishes ; under this last condition the dimensions of V, perpendicular to the plane of x, y become indefinitely small; in other words, the solid degenerates into a plane area, simultaneously the projected curve degenerates into a curve of single curva- ture lying on this plane area. The equation to the primitive solid being given in the form . z=(2,Y), then the superficial area of this solid will be given by the equation Vee Bea SOME PROPERTIES OF PROJECTED SOLIDS. 229 the superficial area of the z derivative will be given by the equation JV elEy org) s.= (4/1 +n (14052 +n eee) dx dy. When n=1, and therefore m=o0, /=0, we have S=8:. When n=0 it will be observed that S, does not vanish ; the equation then assumes the form = (lax dy. i it) At Mag wis Sth! sabigh by ain tothe ay NT yy oer Sm ge mu ‘kee 4 a a F 5: i i i 1: a A ¥ ee 4 in , 8 4 = At aa i Pen w -, in 4 . be : - os . ¥ . r - i i ~% : fi i ae oe a . - bd . . . ‘ 7 wat i 7 Ved By ' y ve , Plate l. gus F T+ ; aan: Dore ne Te tit See 374 Series, Vol. VIL. 2) 4 ” On the Development of the Common Frog « Dr. Alcock: Memoirs, Manchester Literary and Philosophical Society. y S ss ” Z Society. Ss, Manchester Literary and Lhilosophita ae z « On the Development of the Cornmon Frog SES, VOL. VITT. 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REE Pies ar P17; oe n KS eo = ae ae Lr CHAR SATS aa. Ss ; Woe gee Dr. Alcock: On the Development of the Common Frog” Memoirs, Manchester Literary and Philosophical SOCLELY . i oe ee r ee eS > oe PAs a oad de eo Pla. Ih ”" iv pa r Ww ery rope z in Y Vi) pad A BEAN, Cori ba” ‘s i ies & Des f ; Ne ; a Je Nins gir % fina ear | Kl Nah ‘pte « Manchester Literary and Lhilosophival Society. 00 (2) she iiss = oO = = ‘Oo 5) @ ors er) ue 2) a S a) = “On Se] oO > ‘Oo O me) ho sy iS eS x< “) .@) x2) a cs (= 374 Series, Vol. VIL. Memoirs, THE COUNCIL OF THE MANCHESTER LITERARY AND PHILOSOPHICAL SOCIETY. Aprin 29, 1884. Presivent. WILLIAM CRAWFORD WILLIAMSON, LL.D., F.R:S. CWice-Prestvents. HENRY ENFIELD ROSCOE, B.A., LL.D., Pu.D., F.BS., F.C.S. JAMES PRESCOTT JOULE, D.C.L., LL.D., F.R.S., F.CS., Hon. Mem. C.P.S., erc. ROBERT ANGUS SMITH, Pu.D., F.R.S., F.C.S. OSBORNE REYNOLDS, M.A., F.R.S., Proressor or ENGINEERING, Owens CoLLEGE. Secretaries. JOSEPH BAXENDELL, F.R.A.S,, Corr. Mem. Roy. Puys.-Ecoy. Soc. Kontespere, anv Acap. Sc. Lit. Patermo. JAMES BOTTOMLEY, B.A., D.Sc., F.C.S. Treasurer. _ CHARLES BAILEY, F.LS. Pibrarian. FRANCIS NICHOLSON, F.ZS. Other Members of the Council. ROBERT DUKINFIELD DARBISHIRE, B.A., F.G.S. BALFOUR STEWART, LL.D., F.R.S. CARL SCHORLEMMER, F.R.S. WILLIAM HENRY JOHNSON, B.Sc. HENRY WILDE. JAMES COSMO MELVILLI, M.A., F.LS. “HONORARY MEMBERS. DATE OF ELECTION. 1847, Apr. 20. 1843, Apr. 18. 1860, Apr. 17. 1859, Jan. 25. 1866, Oct. 80. 1884, Mar. 18. 1844, Apr. 30. 1869, Mar. 9. 1843, Feb. 7. 1858, Apr. 19. 1848, Jan. 25, Adams, John Couch, F.R.S., V.P.R.A.S., F.C.P.S., Lowndsean Prof. of Astron. and Geom. in the Univ. of Cambridge. The Observatory, Cambridge. Airy, Sir George Biddell, K.C.B., M.A., D.C.L., F.RB.S., V.P.R.A.S., Hon. Mem. R.S.E., R.LA., M.C.PS., &e. TheWihite House, Croom’s Hill, Greenwich Park, SE. Bunsen, Robert Wilhelm, Ph.D., For. Mem. RB.S., Prof. of Chemistry at the Univ. of Heidelberg. Heidelberg. Cayley, Arthur, M.A., F.R.S., FLR.A.S. Garden House, Cambridge. Clifton, Robert Bellamy, M.A., F.R.S., F.R.A.S., Professor of Natural Edenh, Oxford. New Museum, Oxford. abe Dancer, John Benjamin, F.R.A.S. Manchester. Dumas, Jean Baptiste, Gr. Off. Legion of Honour, For. Mem. R.S., Mem. Imper. Instit. France, &e. 42 Rue Grenelle, St. Germain, Paris, Frankland, Edward, Ph.D., F.R.S., Prof. of Chemistry in the Royal School of Mines, Mem. Inst. Imp. (Acad. Sci.) Par.,&c. The Yews, Reigate Hill, Rei- gate. Frisiani, nobile Paolo, Prof., late Astron. at the Ob- serv. of Brera, Milan, Mem. Imper. Roy. Instit. of Lombardy, Milan, and Ital. Soc.Sc. Milan. Hartnup, John, F.R.A.S. Odservatory, Liverpool. Hind, John Russell, F.R.S., F.R.A.S., Superintendent of the Nautical Almanack. 3 DATE OF ELECTION. 1866, Jan. 23, 1869, Jan. 12. 1872, Apr. 30. 1852, Oct. 16. 1844, Apr. 30. 1851, Apr. 29. 1866, Jan. 23. 1866, Jan. 23. 1849, Jan. 23. 1872, Apr. 30. 1869, Dec. 14. 1851, Apr. 29. 1861, Jan. 22. 1868, Apr. 28. Hofmann, A. W., LL.D., Ph.D., F.RB.S., F.C.S., Ord. Leg. Hon. S™™ Lazar. et Maurit. Ital. Eq., &c. 10 Dorotheenstrasse, Berlin. Huggins, William, F.R.S., F.R.A.S. Upper Tulse Lfhill, Brixton, London, S.W. Huxley, Thomas Henry, LL.D.(Edin.), Ph.D.,Pres.B.5., Professor of Natural History in the Royal School of Mines, South Kensington Museum, F.G.S., F.Z.S., F.L.S., &c. School of Mines, South Kensington Museum, S.W., and 4 Marlborough-place, Abbey- road, N.W. Kirkman, Rey. Thomas Penyngton, M.A., F.R.S. Croft Rectory, near Warrington. Owen, Sir Richard, K.C.B., M.D., LL.D., F.RS., F.L.S., F.G.S., V.P.Z.S., F.R.C.S. Ireland, Hon. M.R.S.E., For. Assoc. Imper. Instit. France, &c. Sheen Lodge, Richmond. Playfair, Rt. Hon. Lyon, C.B., Ph.D., F.R.S., F.G.S., M.P., F.C.8., &e. 68 Onslow-gardens, London, S.W. Prestwich, Joseph, F.R.S., F.G.S. Shoreham, near Sevenoaks. Ramsay, Sir Andrew Crombie, F’.R.8., F.G.S., Director of the Geological Survey of Great Britain, Professor of Geology, Royal School of Mines, &c. Greological Survey Office, Jermyn-street, London, S.W. Rawson, Robert, F.R.A.S. Havant, Hants, Sachs, Julius, Ph.D. Prague. Sorby, Henry Clifton, F.R.S., F.G.S., &e. Broomfield, Sheffield. Stokes, George Gabriel, M.A., D.C.L., Sec. B.S., Lucasian Professor of Mathem. Univ. Cambridge, F.C.P.8., &c. Lensfield Cottage, Cambridge. Sylvester, James Joseph, M.A., F.R.S., Professor of Mathematics. Johns Hopkins University, Baltwmore, Tait, Peter Guthrie, M.A., F.R.S.E., &c. Professor of Natural Philosophy, Edinburgh. 4. DATE OF ELECTION. 1851, Apr. 22. 1872, Apr. 30. 1868, Apr. 28. Thomson, Sir William, M.A., D.C.L., LL.D., F.R.SS. L. and E., For. Assoc. Imper. Instit. France, Prof. of Nat. Philos. Univ. Glasgow. 2 College, Glasgow. Trécul, A., Member of the Institute of France. Paris, Tyndall, John, LL.D., F.R.S., F.C.S., Professor of Natural Philosophy in the Royal Institution and Royal School of Mines. Royal Institution, London, W. CORRESPONDING MEMBERS. 1860, Apr. 17. 1861, Jan. 22. 1867, Feb. 5. 1870, Mar. 8 1866, Jan. 23, 1861, Apr. 2. 1849, Apr. 17. 1850, Apr. 80. 1882, Noy. 14. 1862, Jan. 7. 1859, Jan. 25. Ainsworth, Thomas. Cleator Mills, near Egremont, Whitehaven. Buckland, George, Professor, University College, Toronto. Zoronto. Cialdi, Alessandro, Commander, &c. Rome. Cockle, The Hon. Sir James, M.A., F.R.S., F.R.AAS., F.C.P.8. Sandringham-gardens, Ealing. De Caligny, Anatole, Marquis, Corresp. Mem. Acadd. Se. Turin and Caen, Soce. Agr. Lyons, Sci. Cher- bourg, Liége, &e. Durand-Fardel, Max, M.D., Chev. of the Legion of Honour, &c. 386 Rue de Lille, Paris. Girardin, J., Off. Legion of Honour, Corr. Mem. Im- per. Instit. France, &e. Lille. Harley, Rev. Robert, F.R.S., F.R.A.S. College Place, Huddersfield. Herford, Rev. Brooke. Arlington-street Church, Bos- ton, US. Lancia di Brolo, Federico, Duc., Inspector of Studies, &e. Palermo. Le Jolis, Auguste-Frangois, Ph.D., Archiviste per- pétuel and late President of the Imper. Soc. Nat. Se. Cherbourg, &c. Cherbourg. 5 DATE OF ELECTION. 1857, Jan, 27, 1862, Jan. 7. 1869, Jan, 12. 1867, Feb. 5. 1834, Jan, 24, 1881, Jan. 11. 1861, Jan. 22. 1873; dan." 7; 1870, Dec. 13. 1861, Jan. 22. 1882, Jan. 24. 1837, Aug. 11. 1881, Nov. 1. 1874, Nov. 3. 1865, Nov. 15. 1883, Oct. 16. 1876, Nov. 28. 1867, Noy. 12. 1858, Jan. 26. 1847, Jan. 26. 1877, Nov. 27. Lowe, Edward Joseph, F.R.S., F.R.AS., F.GS., Mem. Brit. Met. Soc., &e. Nottingham. Nasmyth, James, C.E., F.R.A.S., &c. Penshurst, Tun- bridge. Saint Venant, Barré de, Ingénieur en chef des Ponts et Chaussées, Corr. Soc. Sci. de Lille et de l’Acad. Romaine des Nuovi Lincet, &e. Schonfeld, Edward, Ph.D., Director of the Mannheim Observatory. Watson, Henry Hough. Bolton, Lancashire. ORDINARY MEMBERS. Adamson, Daniel, F.G.S. The Towers, Didsbury. Alcock, Thomas, M.D., Extr. L.R.C.P. Lond., M.R.C.S. Engl. L.S.A. Oakfield, Ashton-on-Mersey. Allman, Julius, 2 Vectoria-street. Angell, John. Manchester Grammar School. Anson, Ven. Archd. George Henry Greville, M.A Birch Rectory, Rusholme. Arnold, William Thomas, M.A. Phymenth Grove. Ashton, Thomas. 386 Charlotte-street. Ashton, Thomas Gair, B.A. 36 Charlotte-street. Axon, William E. A., M.R.S.L., F.S.L. Fern Bank, 1 Bowker-street, Higher Broughton. Bailey, Charles, F.L.8. Ashfield, College-road, Whalley Range. Baker, Harry, F.C.S. 262 Plymouth-grove. Barratt, Walter Edward. -Kersal, Higher Broughton. Barrow, John. Stafford-road, Ellesmere-park, Eccles. Baxendell, Joseph, F.R.A.S., Corr. Mem. Roy. Phys. Econ. Soc. Konigsberg, and Acad. Sc. & Lit. Palermo, 14 Liverpool-road, Birkdale, Southport. Bazley, Sir Thomas, Bart. Riversleigh, Lytham. Becker, Wilfred, BAA. 4 Commercial Buildings, Cross-street. 6 DATE OF ELECTION. 1878, Nov. 26. Bedson, Peter Philips, D.Sc. College of Science, New- castle-upon- Tyne. 1847, Jan. 26. Bell, William. 51 King-street. 1870, Nov. 1868, Dee. 1861, Jan. 1875, Noy. 1855, Apr. 1861, Apr. 1844, Jan. 1860, Jan. 1846, Jan. 1872, Nov. 1874, Dec. 1842, Jan. 1854, Apr. 1841, Apr. 1853, Jan. 1859, Jan. 1861, Nov. 1848, Jan. 1876, Apr. 1861, Apr. 1854, Feb. 1871, Nov. 1853, Apr. 1878, Nov. 1869, Nov. 1861, Dee. 15. 15. 22. 10. Bennion, John A., B.A., F.R.A.S. Barr Hill Mount, Bolton-road, Pendleton. Bickham, Spencer H.,jun. Endsleigh, Alderley Edge. Bottomley, James, D.Sc., B.A., F.C.S. 7 Avenue, Lower Broughton. Boyd, John. 58 Parsonage-road, Withington. Brockbank, William, F.G.S. Prince’s Chambers, 26 Pall Mall. Brogden, Henry, F.G.S. Hale Lodge, Altrincham. Brooks, William Cunliffe, M.A., M.P. Bank, 92 King-street. Brothers, Alfred, F.R.A.S. 14 St.-Ann’s-square. Browne, Henry, M.D.,M.A., M.R.C.S. Engl. Wood- hey, Heaton Mersey. Burghardt, Charles Anthony, Ph.D. 110 King-street. Carrick, Joseph. Prince’s Chambers, 26 Pall Mail. Charlewood, Henry. 1 St. James’s-square. Christie, Richard Copley, M.A., Chancellor of the Diocese. 2 St. James’s-square. Clay, Charles, M.D., Extr. L.R.C.P. Lond., L.R.C.S. Edin. 101 Piccadilly. Cottam, Samuel, F.R.A.S. 6 Essex-street. Coward, Edward. Heaton Mersey, near Manchester. Coward, Thomas. Bowdon. Crowther, Joseph Stretch. Endsleigh, Alderley Edge. Cunliffe, Robert Ellis. 18 Vine-grove, Pendleton. Cunningham, William Alexander. Auchenheath Cot- tage, Lesmahagow, by Lanark, N.B. Dale, John, F.C.S. Chester-road. Dale, Richard Samuel, B.A. Cornbrook Chemical Works, Chester-road. Darbishire, Robert Dukinfield, B.A., F.G.S. 26 George-street. a7 Davis, Joseph. Engineer's Office, Lancashire and Yorkshire Railway, Hunt’s Bank. Dawkins, William Boyd, M.A., F.R.S. Owens College. Deane, William King. 25 George-street, Cornbrook Chemical Works, Museum, 7 DATE OF ELECTION. 1879, Mar. 18. Dent, Hastings Charles. 20 Thurloe-square, London, 1883, Oct. 2. 1840, Jan. 21. 1881, Noy. 1. 1874, Nov. 3. 1875, Feb. 9. 1878, Apr. 30. 1862, Nov. 4. 1839, Jan. 22. 1873, Dec. 16. 1828, Oct. 31. 1833, Apr. 26. 1864, Mar, 22. 1881, Noy. 1. 1851, Apr. 29. 1884, Jan. 8. 1846, Jan. 27. 1882, Oct. 17. 1884, Jan. 8. 1873, Dec. 2. 1884, Jan. 8. 1872, Feb. 6. 1870, Nov. 1. 1878, Nov. 26. 1848, Apr. 18. 1842, Jan. 25. SW. Faraday, Frederick James, F.L.S. Brazennose Club. Gaskell, Rev. William, M.A. 46 Plymouth Grove. Greg, Arthur. Hagley, near Bolton. Grimshaw, Harry, F.C.S. Zhornton View, Clayton. Gwyther, R. F., M.A., Lecturer on Mathematics, Owens College. Owens College. Harland, William Dugdale, F.C.S. 25 Acomb-street, Greenheys. Hart, Peter. 192 London-road. Hawkshaw, Sir John, F.R.S., F.G.S., Mem. Inst, C.E. 33 Great George-street, Westminster, London, SW. ‘ Heelis, James. 71 Princess-street. Henry, William Charles, M.D., F.R.S. street, Lower Mosley-street. Heywood, James, F.R.S., F.G.8., F.S.A. 26 Ken- sington-Palace- Gardens, London, W. Heywood, Oliver. Bank, St. Ann’s-street. Higgin, Alfred James. 22 Little Peter-street, Gay- thorn. Higein, James. 11 £ast- Inttle Peter-street, Gaythorn. Hodgkinson, Alexander, M.B., D.Sc. Claremont, Higher Broughton. Holden, James Platt. 38 Temple Bank, Smedley Lane, Cheetham. Holt, Henry. Fairlea, Didsbury. Hopkinson, Charles. 29 Princess-street. Howorth, Henry H., F.S.A. Derby House, Eccles. Hurst, Charles Herbert. Owens Colleye. Jewsbury, Sidney. 389 Princess-street. Johnson, William H., B.Sc. 26 Lever-street. Jones, Francis, F.R.S.E., F.C.S. Grammar School. Joule, Benjamin St. John Baptist. 12 Wardle-road, Sale. Joule, James Prescott, D.C.L., LL.D., F.R.S., F.C.S., Hon. Mem. C.P.S., and Inst. Eng. Scot., Corr. Mem. Inst. Fr. (Acad. Se.) Paris, and Roy. Acad. Se. Turin. 12 Wardle-road, Sale. s 8 DATE OF ELECTION, 1852, Jan. 1862, Apr. 1884, Jan. 1863, Dec. 1850, Apr. 1884, Jan. 1857, Jan. 1870, Apr. 1850, Apr. 1866, Noy. 1859, Jan. 1875, Jan. 1879, Dec. 1873, Nov. 1864, Noy. 1873, Mar. 1879, Dec. 1881, Oct. 1877, Nov 1861, Oct. 1850, Jan. 1873, Mar. 1862, Dec. 1861, Jan. 1844, Apr. 1861, Apr. 27. 29. 27. 19. 30. 2. 30. 30. 1876, Nov. 28. 1881, Nov. 29. 8. 16. 30. 22. 13. 25. 26. 4, i 1s. Kennedy, John Lawson. 47 Mosley-street. Knowles, Andrew. Swinton Old Hall, Swinton. Larmuth, Leopold. Owens College. Leake, Robert, M.P. 75 Princess-street. Leese, Joseph. Fallowfield. London, Rev. Herbert, M.A. The Rectory, High Leigh, Cheshire. Longridge, Robert Bentink. Yew-Tree House, Tabley, Knutsford. Lowe, Charles. 48 Piccadilly. Lund, Edward, F.R.C.S. Eng., L.S.A, 22 St. John- street, McDougall, Arthur. City Flour Mills, Poland-street, Maclure, John William, F.R.G.S. Cross-street. Mann, John Dixon, M.D., M.R.C.P. Lond. 16 Sz. John-street. Marshall, Alfred Milnes, M.A., D.Sc., Professor of Zoology, Owens College. Owens College. Marshall, Rev. William, B.A. 81 High-street, Chorl- ton-on-Medlock. Mather, William. Iron Works, Deal-street, Brown- street, Salford. Melvill, James Cosmo, M.A., F.L.S. Kersal Cottage, Prestwich. Millar, John Bell, B.E., Assistant Lecturer in Engi- neering, Owens College. Owens College. Mond, Ludwig. Wannington Hall, Northwich. Moore, Samuel. 25 Dover-street, Chorlton-on-Medlock. Morgan, John Edward, M.B., M.A., M.R.C.P. Lond., F.R. Med. and Chir. 8. 1 St. Peter’s-square. Newall, Henry. Hare Hill, Inttleborough. Nicholson, Francis, F.Z.S. 62 Fountain-street. Ogden, Samuel. 10 Mosley-street West. O’Neill, Charles, F.C.S., Corr. Mem. Ind. Soc. Mul- house. 72 Denmark-road. Ormerod, Henry Mere, F.G.S. 5 Clarence-street. Parlane, James. Rusholme. Parry, Thomas. G'rafton-place, Ashton-under-Lyne, Peacock, Richard. Gorton Hall, Manchester. 9 DATE OF ELECTION. 1874, Jan. 1854, Jan. 1861, Jan. 1854, Feb. 1859, Apr. 1869, Noy. 1883, Apr. 1880, Mar. 1860, Jan. 1864, Dec. 1822, Jan. 1858, Jan. 1851, Apr. 1870, Dec. 1842, Jan. 1873, Nov 1881, Nov 1852, Apr. 1876, Nov. 1844, Apr. 1884, Jan. 1859, Jan. 1870, Nov. 1863, Oct. 1884, Jan. 1884, Mar. 1873, Apr. 1860, Apr. 13. 24, 29. 13. 25. lS 29. 20. 28. 29. bo ten