414 Mr DE morgan, ON THE THEORY OF 



J ^'2""'^? «""+'* da? = (2k + 1) (2k + 2) ... (2k + 2m) J <px . s^dx. 



Dismissing for a while the idea of the number of quantities being infinite, I now ask 



what is the law of facility of value which gives A^^^ 1 . S .5...2A; - 1 . ^*, for all integer 



values of h. We know one such law, which has the modulus '\/ c . q'"^ : y/ -n : this, c 



being (sJa)"'? satisfies all the conditions. 



^\ 

 If we could determine a function V, for which / Vi^dx = from k = upwards, for all 



integer values, we might add this function to the modulus already obtained. We might 

 almost deduce, a priori, that though such functions could be found if we could dispense with 



Vdx = 0, the necessity of this condition is an insuperable barrier. This 



discussion will, however, be rendered unnecessary by the following mode of proceeding. 



If to the quantity whose modulus is in question we add a constant, the character of that 

 modulus is unaltered: (px being the modulus, all we have to say is that (pxdx now represents 

 the probability of the value lying between const. + x and const. + x + dx. If we add a, 

 quantity of variable value ^, of indefinite modulus xj/x, we may, at the close of our investigation, 



so change ^l/x that every case of / x^/x x^dx shall diminish without limit, from k=l upwards. 



We suppose \l/x to be an even function. The modulus of the sum, ^ + a?, as we shall pre- 



sently see, is / y^(q— x) <pxdx, which, multiplied by dq, represents the probability of the 



sum lying between q and q + dq. Expanding ■\\i(q-x) by Taylor's theorem, and paying 

 attention to preceding conditions, we have for this modulus, \c being (2A^~^\, 



>j/q + ^a ^ + 1 ■ SAI ^^—!- + 1.3.5^^ ^ J/^ ^ + ... 

 ^^ 2 ^2.3.4 ^2..S.4.5.6 



= - V- f V"' {^(q+x^ + y\,(q- x)} dx = \ sj - f e""' {x// (x + q) + ^ (a, - q)} dx 



= Ay'-fe-'^'(^x + ^"x^+f"'x^^ + ...)dx. 



Expand e""* in powers of x, make the multiplication, and we have as many terms as there 

 \i/'^"'a;.a?^*da?. The integrations already described shew that the only terms 

 independent of yj/x are those which arise from the cases of 



(- 1)*— ^ ^ fyL^"'^xx"'dx, which is '^ ^ ? ; 



'^ ^ 2.3...k 2.3...2kJ ^ 2, 3. ..ft 



