the Inversion of the Independent Variable. 537 



J Thus_, ex. gr. 



^"^^ ^^-"1.2x1. 2. 3. 4. 5. 6"-^^ H1 



^^'^^-1.2.3x1.2.3.4.5 = ^^ 



(4 4) - 1 . 1>^»3.4.5.6.7.8 

 ^'^^"2 1. 2. 3. 4x1. 2. 3. 4-^^ 



^2 9 n^-l 1.2.3.4.5.6.7.8.9 

 ^ ' '^ "2 ' 1 .2x1.2x1.2.3.4.5 

 ^9^ ^A^ l ■^■3.4.5.6.7.8 .9 

 ^'''^^-l. 2x1. 2. 3x1. 2. 3. 4 

 ^/q q ON 1 1.2.3.4.5.6.7.8. 9 



^o,o, ^;- 3 3 3 • 1.2.3x1.2.3x1.2.3 



rs 9 9 IN- 1 1.2.3.4.5.6.7.8.9.10 



^'''^" 1.2. 3 '1.2x1. 2x1. 2x1. 2. 3. 4 



(9 9^^\--L. _L 3 .2.3.4.5.6.7.8 .9.10 

 ^ ' ' ' ^ - 1 . 2 ^ 1 . 2 ^ 1.2x1.2x1.2.3x1.2.3' 

 and so on. The general law is obvious ; and to prove its appli- 

 cability in general, we bave only to show that if it be true for 



the case of -j^j it is true for _ — —. The proof is as follows. 

 dx"" dx"""^ 



Let in general [/, m, n, &c.] indicate the value of 

 1.2.3 (/+W + 71 + &C.) 



1.2.../xl.2...mxl.2. ..wx&c' 

 without reference to /, m, w, &c. being equal or unequal inter se. 

 Lemma 1. It is very easily seen that 



^^ [/, m, n, &c.] = [/— '1, m, n, &c.] + [/, m — \, n, &c.] 



+ [/, m, n — \, &c.] -f-&c. 

 If now we use the notation [r : ^, s : o-, t '.r, &c.] as an abbre- 

 viated form of the notation [p, p, p . . . to r terms, a, a, a ... to 

 s terms, r, t . . . to / terms, &c.], it is obvious that the equation 

 last written becomes 



\r:p,s:(ryt:T, &c.] =7* x [p — \, (^ — 1) : p, s:<T,t:r, &c.] 



+s[r:/9,o— 1, {s—\)'.<T, ^:t,&c.] ^-/[r:p, s:cr,T— 1,(/— l)r,&c.]+&c. 



Lemma 2. Let C(r:p, s:a, tir, &c.) denote the number of ways 

 in which r . p-\-s . cr-\-t .t, &c. can be taken in combinations of 

 p, p, . . . tor places, a, a, ... to s places, &c., then upon the sup- 

 position that p, cr, T, &c., which are to be understood as arranged 

 in an ascending order of magnitude, are all unequal, we shall 

 have (using for shortness G(r) to denote 1 . 2 . 3 . . . r), 



Phil. Mag. S. 4. No. 55. Suppl. Vol. 8. 2 N 



