12 MR. E. CUNNINGHAM ON THE NORMAL SERIES 



These equations manifestly determine x^ except for its first and second elements, 

 the second being known as soon as p l is. 



We are also at once faced with a condition necessary for the possibility of the 

 solution under the assumptions made as to the form of x, viz. : 



/' = 0. 



This condition arises from the e," 1 equation of the set, and as, in the ensuing 

 discussion, the e^' 1 equation of each set is most important we shall here introduce a 

 notation for it, viz., X r will stand for the e/ 11 equation of the (7-+l) th set; i.e., of the 



set 



(a f + l -0 l p + l )x r +... = 0. 



This equation will not contain any element of x r . 



A similar notation will be adopted for the equations Y, Z... for the coefficients in 

 the other columns of TJ. 



If the second element of the first row of X P +\ t e C 2\, the equations Y are 



(^-H-tf'W l)2/0-^0 = 0, 



(a.j,+i-8 a 1 ,+i)yi+(a. f -0 p )y -c lsl x 1 = 0, 



Of these the first is satisfied by 



provided we take c 21 = 1 = corresponding element in a p+l . 



Considering each of the columns in succession we have thus, with i) = 1. 

 XP+I = a p+\- 



The second of the equations Y gives 



-X 1 1 = 0, 



-V-xf = o, 



atp 8 "'- fl^* = 0, 



(j^i-/'i)2/ 1 ei+1 + <' +1 -*i ei+1 = 0, 



which, when x l is known, determine y l save for its first element, and its third until 

 OP is known. 



The exceptional equation Y,, af'-xS = 0, gives us again a necessary condition for 

 the possibility of the solution in view, a/^' + a'- 1 = 0, 



