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THE 

ASSUEANCE MAGAZINE, 

AND 

JOURNAL 

OF THE 

INSTITUTE OF ACTUARIES. 



On the Improvement of Life Contingency Calculation. By Edwin 
James Farren, Esq., one of the Vice Presidents of the Institute 
of Actuaries. 

[Read before the Institute 8th January, 1855, and ordered by the Council to 

be printed.] 

J.HE prevailing system of life contingency calculation is one not 
of variable but of invariable quantities. At the very threshold the 
admission of two such important assumptions is asked for, as that 
•the rate of mortality is always invariable at the same age, whether 
old or young, and that the rate of interest is equally invariable for 
all periods, whether long or short. Upon these admissions of in- 
variability a system is formed for assessing the relative values of 
different cases, thereby necessarily in every instance indicating an 
invariable answer; and with such indications the system rests 
content. Whether such assessments, however logically fair in 
connection with agreed postulates of invariability, are themselves 
eventually justified by the same invariability of actual result as was 
a priori assumed, has not hitherto been commonly brought within 
the general scope of the actuary's studies. Directly, however, he 
is called on to take upon himself the practical responsibility of 
upholding this theory of invariability, he is somewhat surprised to 
find that, good as the mere logic of his studies may have been, it 
is by no means an easy task to connect such logic with the nature 
of the events he may see passing around him. In place of a 
definite rate of mortality at each age, he may find a perfect series 
vol. v. o 



186 On the Improvement of [April 

of such rates — in place' of one uniform rate of interest, he may 
find one portion of the funds yielding no interest whatever, while 
other portions may he lent upon classes of security, fruitful indeed 
in interest, but not as equally fruitful in obvious expedients for 
realization of the principal, should such be desirable. The con- 
viction is thus forced upon him at a very early period of his career, 
that, as a practical man, he must either henceforth shut his eyes 
altogether to the prevailing theory of the subject, or else open his 
eyes much wider than as a student he was taught to expect would 
be necessary in the application of rules professedly based upon 
invariable elements. 

I am aware it may be said, that although the rates both of 
mortality and interest, as used by actuaries under the prevailing 
system, are certainly specific rates, yet that they are to be con- 
sidered as averages, and therefore typical, as well in theory as in 
practice, of a diversity of rates. But I would ask, has the subject 
ever been so treated? Can we find, for instance, in the treatises 
of those who are termed by common consent, and therefore rightly 
termed, our standard writers, not merely a single chapter, but even 
a single page, on the proper calculation and employment of average 
results, as applicable to insurance transactions? So long then as 
this omission exists, so long I think we may without presumption 
assert that the prevailing system of life contingency calculation is 
susceptible of considerable improvement. But in alluding to this 
omission, I particularly wish to guard myself from the inference 
that I thereby desire to stigmatize the labours of such men as 
Price, Morgan, Baily, and Milne, or of our late excellent cotem- 
porary David Jones, as not worthy of the favourable reception they 
have received. On the contrary, I believe (and it is one of the 
objects of the present paper to illustrate such belief) that the style 
of treatment adopted by these pioneers of our literature constituted 
the only style that was really applicable to the nature of insurance, 
as then publicly understood. Indeed, in all branches of natural 
philosophy we find a similar assumption of invariability at an early 
period of their history, as preparatory to the more comprehensive 
study of deviations required by modern science. One of these 
branches presents us with so close a parallelism to the line of pro- 
gress I wish to illustrate as advisable in insurance calculations, 
that I cannot consider it wholly a digression to allude to it even in 
some detail. 

In the early history of navigation, we find it taken almost as 
the basis of the science that the compass needle pointed in a fixed 



1855.] Life Contingency Calculation. 187 

direction, and that such direction was due north. The utility of so 
simple an assumption in early days can scarcely be overrated; the 
more especially as it was by this simplicity then rendering itself 
acceptable that the real properties of the magnet have been reserved 
for more elaborate study in our own time. Indeed, had variability 
in place of invariability been originally taken for its prevailing 
attribute, the important uses of the mariner's compass would 
probably have been lost to modern science, inasmuch as our sim- 
plicity-loving ancestors might have considered such indications too 
uncertain and complex either to be useful to themselves or to their 
posterity. In further illustration of this, we find that for a long 
time the simplicity of such a theory of invariability completely 
overrode the nature of the actual facts. Gradually, however, the 
proverbial stubbornness of facts developed itself in this as in other 
sciences, and brought about the admission that the compass needle 
did not really and in fact point due north. There was as yet, 
notwithstanding, no absolute surrender of the principle of fixity of 
direction itself; such would have been too sudden a change either 
to have been expected or even desired. A variety of adjustments 
were accordingly advocated, each claiming some favourite point as 
that of the real fixity. The correct theory, nevertheless (started, I 
believe, by Gunter, and now admitted by all conversant with the 
subject, because the only theory that can interpret all the facts), is, 
that the direction of the compass needle is variable even in the 
same localities, and must be so apprehended by mariners, if safety 
based upon truth, and not -merely upon simplicity of theory, be 
their aim. It is, then, by the known variability of the compass 
needle, and not by a pseudo invariability, that the triumphs of 
modern over ancient navigation have been achieved. 

The use I seek to make of this as a parallelism in illustration 
of the subject before us is, I presume, sufficiently obvious. I con- 
sider the formerly universal adoption of the Northampton Table 
and 3 per cent, as typified by the assumption that the compass 
needle pointed fixedly due north, and the various petty controversies 
for other fixed points as equally typified by the various pros and 
cons for the Carlisle and other tables. Further, that the true 
theory, in this as in the former subject, is one strictly of variation, 
both as to mortality and interest, and that it must be so accepted 
as the only guide to safe practice, if we would avoid those rocks 
and shoals which a purblind adherence to a fixed in place of a 
variable course might unpreparedly develope. Indeed, the distinc- 
tion between the proper treatment of variable and invariable ele- 

o 2 



188 On the Improvement of [April 

ments is precisely the distinction that characterizes the vocation of 
an actuary as compared with that of an accountant. Thus the 
actuary who should take probabilities, because fairly assessed now, 
as necessary certainties hereafter, would be virtually an accountant, 
because he allows no range for the possibly conflicting evidences of 
the future. The accountant, on the other hand, who endeavours 
to put estimates upon fluctuating things to come, is virtually striv- 
ing to be an actuary ; for he cannot but allow that no estimate of 
to-day, unless professedly subject to variation, can pretend to also 
fulfil the condition of being an equally good estimate for a change- 
able to-morrow. In some Offices, I believe, this distinction between 
fixed and variable estimates is already sufficiently carried out — in 
the first case, by the actual amount of assets at one period, as 
compared with the actual amount at another, being illustrated by 
Dr. and Cr. after the manner of accountants ; and in the second, 
by the difference between the amount of the life valuation at one 
period and another being substantiated, actuary-wise, by taking 
into consideration the accrued contingencies of the past as compared 
with the range of contingencies to be provided against for the 
future. The actuaries, then, of what I shall venture to call the 
old school, were essentially accountants in the modern sense, for it 
was only with fixed quantities they professed to deal, as is suffi- 
ciently proved by their assuming a fixity when they found it not. 

The results of such a system have been exactly those to be 
expected. Where exorbitancy secretly existed, as in the rates 
required by the Insurance Societies, there the errors of a fixed and 
affected precision eventually came to light, in the shape of bonuses 
added to the sums assured, in varying amounts from time to time, 
in strange contrast with the declared formality of the original 
fixedness of calculation. Where no such exorbitancy of charge 
was allowable or even possible, and the fixed calculations had to 
stand or fall by their own merits, there the dangers of professing 
to deal with variable quantities as if they were absolutely invariable 
were unfortunately not so easily neutralized. To the numerous 
Friendly and Annuity Societies of the last century throughout the 
country, a reliance on fixed tables and on such tables alone has at 
once proved a delusion and a snare ; for it tempted them to appro- 
priate the temporary surplus of a day, in the vain expectation that 
the fixed nature of the tabular values assigned to the future would 
necessarily be sufficient guarantee for the fulfilment of impending 
engagements. 

The nature, then, of the improvement I seek to introduce into 



1855.] Life Contingency Calculation. 18& 

life contingency calculation is to openly take as our guide not 
merely a calculus of averages, but of their fluctuations; and to 
thereby declaredly characterize our methods, not as composing a 
system of specific and precise results, whatever it may be of prices, 
but of results expected to vary between limits of assigned ranges 
of probability. By such a declaration it would at once become 
manifest that our expected gain by computation would not be to 
find even averages themselves invariable, but that their fluctuations, 
being considerably less, would therefore be more readily dealt with 
than the fluctuations of the elements of which they may be com- 
posed. The phrases therefore of the prevailing system implying 
" a true table," or " a true rate of interest," would under such a 
calculus have to give way to average tables, with their probable 
limits and the per centages of their expected deviations. 

What experienced actuary, for instance, can read without feel- 
ing the truth of the following reflection, extracted from the article 
on " Probability" in the Encyclopedia Metropolitana : — 

" Not being well able to decide upon the relative importance of small 
details, calculators on this subject (life contingencies) have hitherto judici- 
ously presented their results such as they ought to be if the tables were 
mathematically exact, and to the nearest farthing. But more extended 
views on the subject of probabilities, and on the nature of observations in 
general, would have caused the time which has been wasted in carrying out 
annuities to many more decimals than the data are good for, to be employed 
in apportioning the risks of fluctuation by estimation of the mean risks of 
the tables." 

Or the following, from the equally excellent article on "Proba- 
bility" in the Encyclopedia Britannica : — 

" We may remark that, although English writers have 'almost without 
exception confined themselves to the explanation of the methods of com- 
puting annuity tables and of determining from them the values of sums 
depending on life contingencies, the aid which this branch of economy 
derives from the general theory of probabilities is by no means confined to 
the consideration of such elementary questions. The number of observa- 
tions necessary to inspire confidence in the tables, the extent to which 
risks may be safely undertaken, the comparative weights of different sets of 
observations, and the probable limits of departure from the average results 
of previous observations in a given number of future instances, are all 
questions of the utmost importance, which come within the scope of the 
calculus, and cannot in fact be justly appreciated by any other means." 

In the concluding part of the latter extract we have, indeed, 
the real explanation of the formal and what I may venture to 
call the " wooden" cast that has been given to the subject by our 
standard writers already referred to : for we are to remember that, 



190 On the Improvement of [April 

though complete masters of their art as then understood, yet that 
they were all teachers or disciples of a school and of a day when 
the differential and integral calculus was but little employed by 
English writers on any branch of science. The omission of such 
processes has now, however, become the exception and not the rule. 
Thus, for instance, if we look into any of our modern treatises on 
mechanics, engineering, or navigation, all of them essentially prac- 
tical subjects, we find every aid that the calculus can give or has a 
chance of giving sedulously pressed into the service. By these 
means the great discovery of Newton and Leibnitz is brought 
home — vicariously indeed, but still effectually — to the uses of the 
humblest mechanic, engineer, or mariner, whenever he has to avail 
himself of what can be done for him, by way of previous calcula- 
tion, in guiding him to the simplest and most trustworthy results. 
Indeed, the modern improvements of the Nautical Almanack alone 
form at once a sufficient and striking illustration of what benefit 
can be achieved by the calculus in devising the best forms for 
practical computation. Whatever therefore may have been the 
opinion of our elder school of writers, I think the time has now 
come for our students when, as in other subjects, the more search- 
ing investigations of the calculus should also be brought to openly 
and commonly bear upon that of life contingencies. 

Indeed, without this or some other extra aid, bow is it possible 
for us to intelligibly explain to a modern public those differences 
of results in various Offices, which, when judged by a hypothetical 
standard of invariability, appear rather to proclaim the failure of 
all methods whatever, than to justify the indications of any par- 
ticular one? So long as this diversity remains unexplained by 
having no proper limits assigned to it, so long assuredly may 
any amount of diversity appear justifiable to boards of manage- 
ment, and actuaries continue to be exposed to the risk of having 
their opinions only treated with respect when not obstructive of 
other money arrangements. That the calculus, especially con- 
sidered as a calculus of averages, contains within itself the means 
of dealing with and explaining these diversities, has been too often 
asserted both by continental and native writers on probability, to 
be strengthened by mere reassertion on my part. But as I am not 
aware that any very ready example, in a professional sense, has 
been given of the sort of assistance to be derived by actuaries from 
this calculus, when treated as a calculus of averages, I shall beg 
leave to conclude this paper by offering at least one such illustra- 
tion, hoping it may prove an incentive to other actuaries to look 



1855.] Life Contingency Calculation. 191 

farther into the subject than perhaps they have hitherto done. 
Before, however, giving such an example, I should wish to state 
that I have purposely selected such an one as will show that I by 
no means pretend, as a practical man, that a more general study of 
the differential and integral calculus by actuaries would materially 
alter the external appearance of insurance results and rates, as 
at present accepted by the public. On the contrary, I believe that 
no actuarial theorizing would or ought to induce the public to be 
otherwise than mainly led by their own experience of the past, 
already somewhat extensive, and every day becoming more and 
more patent to themselves. But there is considerable difference, 
in a professional point of view, between venturing on general 
assumptions, however plausible, and the cautious adoption of 
approximations based upon elaborate investigations. Were then 
the calculus capable of no more than pointing out to us convenient 
approximations, and referring us to its own processes for its justi- 
fication of them, it would still, I think, be" an ally obviously well 
worthy of the actuary's seeking. It is to illustrate the calculus 
in this character that has decided the kind of example I have 
selected. 

Example. — A hundred pounds has to be put out at compound 
interest for twenty years, at rates indefinitely fluctuating between 
3 and 4 per cent, per annum. What is the general average of all 
the possible sums, even to infinity, to which the hundred pounds 
may be thus made to amount ? 

Putting this into the form of a definite integral, we have 

which, when »i=100, a=3, 5=100, c=l, and w=20, as in the 

case before us, becomes innl9 ( ^ J = 199'273l, which 

is the general average amount required. 

Having thus determined what would appear as the more recon- 
dite question of the average amount of a sum at fluctuating rates 
of interest, it may be well, in order to show the ductility of the 
calculus when studied as an extensive system of averages, to also 
determine by its means the more simple question of what is the 
average rate of interest between 3 and 4 per cent., so obviously 
determinable by other means as 3£ ? This extra illustration, how- 
ever simple, is considered advisable; because there may be many 
minds, even in our own profession, so framed, that it is only by 



192 On the Improvement of [April 

treating well known examples, having obvious solutions by the 
current methods, that the reliability of any new method of solution 
is considered admissible by them in more difficult cases. To deter- 
mine, then, the average rate of interest between 3 and 4 per cent, 
by means of the calculus, we have to consider the definite integral of 

xdx = ; 



42 — 32 
which, when a =3, becomes — s — =3|; in exact equality with 

the obvious result of sheer mental arithmetic. I have already 
hinted that the first example is purposely chosen as one suscep- 
tible of an easy approximation, and such has just been portrayed : 
for if £100 be invested at 3J per cent, per annum for twenty years 
throughout, it will amount to £198-9789, a close approximation 
to the general average, or £199-2731, as determined by the cal- 
culus. But should we therefore be justified in saying that the 
£100 must necessarily be considered as having to be invested at 
3£ per cent, per annum throughout ? Decidedly not ; and the less 
so, because all the supposed advantages of such a misstatement 
are more readily obtained by adhering to the scientific truth, and 
saying that the proposed calculation, being really one of indefinite 
fluctuation, has been accordingly so dealt with, and the general 
average ascertained to be £199-2731; without, however, guarantee- 
ing either that or any other as the precise result, that experience 
alone can determine. It is moreover manifest, that 3J as a rate of 
interest could not be connected as such with the average amount 
at the end of the term, for it is as obvious by common arithmetic 
as it is by the calculus that the accumulations between 3J and 4, 
considered as fixed quantities, would more than counterbalance 
those between 3 and 3^ : and hence, whether 3£ would have to 
be considered as affording a good or bad approximation is not 
matter for assumption, but for demonstration ; and it is precisely 
these demonstrations that are beyond the reach of the common 
methods. 

The instances I have given will, I think, sufficiently portray, 
so far as isolated instances can do, the nature of the improvement 
I am advocating in life contingency calculations. It will be seen 
that, though I seek to deprive the prevailing system of its pre- 
tensions to an invariability that does not really belong to it, yet 
that at the same time I propose a similar equivalent, by the 
adoption of the calculus and its limits, to that which has already 



1855.] Life Contingency Calculation. 193 

been accepted; in place of a similarly false invariability, in other 
branches of natural philosophy. It is true, indeed, that before 
being exactly adapted to our wants as actuaries, the calculus must 
be moulded into one of averages ; but this is a transformation so 
legitimate, that I consider no better method of studying the cal- 
culus exists even for the more general mathematical student. 

In concluding this paper, I am perfectly well aware it is con- 
sidered by many as dangerous, in an official and commercial sense, 
for any actuary td show he has been studying other books and 
productions than directories, prospectuses, and advertisements ; 
but I trust that a better spirit, is beginning to prevail, and that, 
within the walls of this Institute at least, any advocacy for the 
improvement of the theory of our subject will be immediately seen 
as also implying a desire to improve its practical aspects. Speak- 
ing for myself, I have long considered that the wants of the public 
are daily forcing upon actuaries the investigation of subjects which 
the incompleteness of the prevailing theory renders it too powerless 
to sufficiently grapple with ; and it is the hope of exciting atten- 
tion to this view of the matter that has induced me to offer the 
present paper. Considerable difficulty, as may be imagined from 
its tenor, has been felt in keeping it within due bounds ; for, had 
examples of general limits been chosen, the subject in this form 
appears to be at present so little understood in its practical bear- 
ings, that at least the range of a lengthened essay, if not of a 
volume, might have been required to treat the matter with that 
fulness of illustration which the importance of it demands. It 
may therefore be allowable for me to attempt to reinforce the 
object of so circumscribed a paper by a general declaration on my 
part that, after having devoted considerable attention, and indeed 
some years, to the subject, I feel confident the proposed change 
from an invariable to a variable calculus as the basis of our 
calculations will be beneficial in every respect. We shall thereby 
be able to wholly dismiss the ancient doctrine of chances, with its 
fixed equalities of paper cards, wooden dice, and similar mechanical 
illustrations, and rely upon the more modern doctrine of pro- 
bability, as the science of observation based upon experience. The 
actuarial adaptations of this doctrine, aided by the calculus, will 
assuredly ultimately bring a class of problems involving averages 
and their fluctuations within reach of our solutions which at pre- 
sent are merely statistically guessed at, even by the most experi- 
enced actuary, the most cautious finance minister, or the most 
learned political economist. To improve our own science, more- 



194 On the Improvement of [April 

over, is virtually a step towards the improvement of others, and 
thereby the better helps us to substantiate the claims of our stu- 
dies to those designations of learned and liberal so duly prized by 
other professions. 



Postscript. — The writer is glad to avail himself of the interval 
between the reading and printing of his paper to state, that he does 
not by such paper claim for his views — as might perhaps hastily, 
without this disclaimer, be inferred — the merit of perfect originality 
as regards the proposed improvement of life contingency calculation. 
Lacroix has long since glanced at the differential and integral cal- 
culus as essentially a calculus of averages, and the calculus itself 
has already been often employed in connection with life contin- 
gencies in England, as by De Moivre, Waring, Young, Gompertz, 
Lubbock, De Morgan, Galloway, Edmonds, and indeed by the 
present writer himself, in his last publication on Life Contingency 
Tables. It is necessary further to remark, in the same spirit, that 
even the terms " true table " and " true rate of interest/' though 
commonly used in the prevailing system, have also been frequently 
associated with the notion of a margin for fluctuations, or accom- 
panied with the qualification that it is only by neglecting variations 
that the epithet of" true " becomes allowable, and that, if it be pro- 
posed to include such variations, that modification of the ordinary 
language should ensue. Reference may be made to Mr. Jellicoe's 
paper, in Vol I. p. 172 of the present Journal, for instances of this. 
It is, then, Tather to excite renewed attention to the subject of 
variability, than to propose it as wholly new, that has been the 
writer's real aim ; and he has accordingly treated the matter in the 
preceding paper in that mixed style of pleading and demonstration, 
as appearing to him the most suitable for such a purpose in its 
more general form. As however it may aid the illustrations already 
given in the paper itself, if a tabular form be presented, and may 
also tend to better satisfy many minds to whom tabular forms are 
more acceptable than even the most earnest disquisitions upon 
principles, such a table is now appended. It may be taken as a 
temporary specimen of the proposed improved manner of dealing 
with such subjects — without, however, the writer's wishing such 
table to be understood as having the exact form, even in his own 
opinion, that may ultimately be best adapted for the purpose 
under consideration. 



1855.] 



IAfe Contingency Calculation. 



195 



Table of the Average Amounts of £100 at Fluctuating 
Bates of Interest. 

Average Amounts of £100 at Compound Interest from 1 to 100 

years, at rates fluctuating between the limits ofO and 6 per cent. 

per annum. 

N.B. The maximum rate of interest is taken at 6, rather than at 5, per cent.; because 
5 per cent., if payable by half yearly, quarterly, monthly, or smaller instalments, can be 
made to exceed 5 per cent, per annum. 



Term 




Term 




Term 




Term 




of 


Average 


of 


Average. 


of 


Average 


of 


Average 


Years. 


Amount. 


Years. 


Amount 


Years. 


Amount. 


Years. 


Amount. 


1 


1030000 


26 


235-9473 


51 


631-3104 


76 


1901-0467 


2 


106-1200 


27 


244-7433 


52 


658-4496 


77 


1990-5568 


3 


1093654 


28 


2539303 


53 


686-8833 


78 


2084-5473 


4 


112-7419 


29 


263-5273 


54 


716-6764 


79 


2183-2499 


5 


116-2553 


30 


273-5538 


55 


747-8971 


80 


2286-9085 


6 


119-9120 


31 


284-0306 


56 


780-6170 


81 


2395-7800 


7 


123-7183 


32 


294-9793 


57 


8149117 


82 


25101350 


8 


127-6813 


33 


306-4428 


58 


850-8605 


83 


2630-2580 


9 


131-8080 


34 


318-3851 


59 


888-5470 


84 


2756-4491 


10 


136-1058 


35 


330-8913 


60 


9280588 


85 


2889-0240 


11 


140-5829 


36 


343-9679 


61 


969-4884 


86 


3028-3153 


12 


145-2472 


37 


357-6426 


62 


1012-9330 


87 


3174-6731 


13 


150-1076 


38 


371-9448 


63 


1058-4948 


88 


3328-4664 


14 


155-1731 


39 


386-9049 


64 


1106-2813 


89 


34900835 


15 


160-4533 


40 


402-5553 


65 


1156-4058 


90 


3659-9337 


16 


165-9581 


41 


4189299 


66 


1208-9873 


91 


3838-4479 


17 


171-6981 


42 


436-0641 


67 


1264-1512 


92 


40260800 


18 


177-6842 


43 


453-9955 


68 


13220292 


93 


4223-3082 


19 


183-9280 


44 


472-7634 


69 


1382-7602 


94 


4430-6361 


20 


190-4416 


45 


492-4090 


70 


1446-4903 


95 


4648-5943 


21 


197-2377 


46 


512-9758 


71 


1513-3730 


96 


4877-7418 


22 


204-3297 


47 


534-5094 


72 


1583-5703 


97 


5118-6675 


23 


211-7316 


48 


557-0580 


73 


1657-2523 


98 


5371-9917 


24 


219-4580 


49 


580-6718 


74 


1734-5982 


99 


5638-3681 


25 


227-5245 


50 


605-4040 


75 


1815-7968 


100 


5918-4853 



Example. — The average amount of all the amounts possible, 
even to infinity, to which £100 can be made to accumulate in 
twenty years, at rates of interest fluctuating between and 6 per 
cent, per annum, is £190*4416 ; as may be seen set forth in the 
table opposite to the term of 20 years. 

When it is remembered that money is more likely to remain 
unproductive, or at per cent., for short than for long periods, it is 
obvious that the relative effect of unproductiveness must be more 
operative when considering brief than enlarged cycles of finance. 
The preceding table, by its averages, properly represents this effect 
among others ; and shows that while on the one hand the average 
amount at the end of the first period or year from the original 
times of the deposits may be taken as sufficiently defined by the 
common mean rate of interest between the limits, or in the present 



196 On the Improvement of Life Contingency Calculation. [April 

case by 3 per cent., yet that,- on the other hand, the period of a 
century is allied by its average amount, in connection with the 
same limits, to the amount productive by a uniform rate through- 
out of about 4 per cent. (4'165). To deal with such wide limits 
as from to 6, and with such durations as a century, is obviously 
to strain the calculus to its utmost; but even in this extreme 
state it will be seen to keep closely attendant upon the incidents 
of practical insurance, for it certainly appears consistent with even 
popular justice that, as a matter of calculation, those who may 
remain longest insured should be also rated as those to be re- 
latively assigned the higher ratios in the general appropriation 
of accumulations of interest. 

The table has been virtually calculated by the aid of the same 
definite integral as that given in the paper, viz. — 

in which n varies by units from 1 to 100 ; a varies indefinitely to 
and from a+g, or from to 6; J = 100; m=100; c=l; and 
g = 6. Or, putting such expressions into the form of a rule, we 
have the following extremely simple one whenever is the lower 
limit : — 

Subtract £100 from its amount, when improved for the whole 
term and one year beyond, at the maximum rate of interest con- 
sidered as uniform as by the common tables. Divide the remainder 
by the product of such maximum rate and the number of years 
including the year beyond, and the quotient will be the average 
amount required. 

Example for Twenty Years. — £100, put out at 6 per cent, per 
annum uniform interest for twenty years and one year beyond, will 
amount by the ordinary tables (Smart's) to £339 - 95636; from 
which if the £100 be subtracted, the remainder is £239*95636 • 
which, divided by 6 times 20 and the year beyond, or 126, gives a 
quotient of £190'4416, which is. the average amount required 
between the assigned limits in conformity with the result as given 
by the table in question. 

The reasons for the trustworthiness of the rule can of course 
only be explained by aid of the calculus itself, or by some allied 
process of reasoning which it would be here out of place to 
dilate upon.