514 
ME. B. GOMPEETZ ON THE SCIENCE 
instead of representatives of value, have their valuable service as well as inconvenience 
to be attended to ; though I prefer much the Continental use of a letter when used as 
a characteristic, to be used, as it is in many cases, as a letter underscored by some one 
or more letters, denoting the quantities of which that letter may be the functional 
characteristic ; as, for instance, to write L„ to express the function of a, which may be 
the age of a person of whom there may be the number living. And having thus 
intruded by a digression on the reader’s attention, I will venture to hope that my still 
continuing the digression will be thought to have some interesting excuse for me, as a 
scientific amusement to the reader ; as a person walking for the sake only of healthy 
exercise in a beautiful garden, may find a pleasure and an advantage to notice the 
elegant flowers, and even the noxious weeds which the ground produces. I vdll state 
that mathematicians who have enlightened the world with the most beautiful discoveries 
use notations which are incorrect, often ambiguous, often furtive, and often contra- 
dictory. Thus in the notation of partial fluxions or partial difierentials, I consider 
4^5 &c., ^ as both incorrect and furtive, and that they may mislead ; instead of 
which I use the expressions ^ , &c. The incorrectness of the former has 
X X^" dx 
been in some degree avoided by W aeing, Laplace, and many other mathematicians of 
eminence, by using the expressions &c., which removes some 
part of my objection, but not the whole of it ; but has the disadA'antage of occupying too 
much space, and is furtive, as it steals the excellent use of the parentheses, which have 
been employed for'useful purposes. And now, requesting my reader to pardon me for 
a digression which some readers may consider uncalled for, I will proceed in the path 
in which I hope to lead him with some satisfaction. 
• — • • 
Art. 3. In the equation 'L^=d.g\ , where signifies the number of persons living at 
the age x, the letters d, g in my paper in the ‘ Transactions’ of 1825 express quantities 
apparently nearly constant during a very long period of life — say, for instance, in the 
Carlisle Table of Mortality between the ages 10 and 60 ; but if we commence our limit 
at a dijfferent age, and terminate it at some other far distant age, those quantities appa- 
rently constant would have difierent values of apparent constancy ; and here the pro- 
duct which would appear in the equation by taking ^=0, would be an arbitrary 
quantity depending on the arbitrary value of Lo as a base, expressive of the arbitrary 
number we intend to set out with as the persons living at the age 0 ; and if, which 
is not the case accurately, the formula were universally trjae Lorn birth to any other 
age, d.g would express the number of persons born on which we intended the formula 
to be constructed; for we should have ljQ=d.g\ but as g is not arbitrary, d would be 
the arbitrary quantity, and we might take (Z=l, if we did not object to fractions in the 
number of the births, to set out witji; and in that case the formula would stand 
generally L,=^l ; if it were universally true from birth to any age ; which, as I observed 
