Sir James Cockle on a Differential Criticoid. 441 



n-l 



2. Let P represent the sum of the products of the natural 



r 



numbers 1, 2, 3, . .n — 1 taken rata time. Then 



n n — 1 n—1 

 P = P + 7lP 



r r r—l 



and 



n — 1 »— 1 



P = 2n P = 2n 2ft . . . £ft 



r r—l 



to r factors, each 2 operating on all that follows it. This agrees 

 with Mr. Scott's result (Quart. Journ. of Math. vol. viii. p. 29). 



n-l 



3. Let H represent the sum of all the homogeneous products 



r 



of?- dimensions which can be formed by the first ft — 1 natural 

 numbers and their powers. Then 



n n—1 n—1 n—1 n—1 



H = H + ftH+ft 2 H+.. +n r - 1 ~H.+n r 



r r r—l r— 2 1 



n—1 n—1 n—1 n—1 



= H + ft(H + ftH + .. +n r - 2 H + w r - 1 ) 



v r—l r— 2 1 



n—1 n 



= H+ftH. 



r r+1 



Hence 



W H = 2nH = 5:/iS(ft + l) W H = 2ft2(ft-}-l)..2(ft + r-l), 



r r—l r— 2 



since 



n+r— 2 



H =S(» + r-l). 

 i 



This agrees in substance with Mr. JefFery's theorem (ibid. vol. iv. 

 p. 370), though not with Mr. Scott's statement of it (ibid. vol. 



unless a = \, in which case it will be (Cz-\-C 1 )e z . Again, take 



S+^-^-o » 



Here we may put £=# a -2 and r/=#- a ; and y—x^ is suggested as, and 

 is, a particular integral when 



m(m— a-f-l) = l (e) 



In the fourth line above equation (2) of the paper the second " A " 

 should be replaced by V ; and in art. 13 thereof, at lines 7 and 1 1, for M 

 read /£?, and at its penultimate line, for "2<e" read ce ; and in line 2 

 of art. 15 expunge the external exponent, adding that the elliptic integral 

 is of the first species. In art. 9, line 17 , for "F" read A. The second 

 casura of a terordinal gives log (£ 2 »?). 



