CuLVERWELL — Solutwns in the Calculus of Variations. 38-3 



"where the suffices 1 and 2 at the left-hand side of Y refer to the 

 variable with regard to which the Y functions, whose formation is 

 explained in § 1, are obtained. 



According to the conditions (8), the terms at the P limits all 

 vanish. 



§ 10. With regard to the terms at the £ limits, the fluxions of 

 8"y are not all arbitrary, because the condition that both AF and BP 

 are to have contact of the proper order with A£ enables us to express 

 some of them in terms of Px and the functions y and y. 



For, if z represent any fluxion of i/i up to yi^°i~^\ inclusive, or of 

 2/2 U-p to y3^°2~^\ inclusive ; and if Z represent the corresponding Y 

 function, we have, by the conditions of contiauity in § 4, because y 

 and y + 8y must have contact of the proper order at A. 



/ 3 = / (z + Sz), or / Sz=l (s-2), (12) 



and 



/B_ IB IB IB _ 



/ z = l {z + h'z), or / h'z = l iz-z); (13) 



from which 



/ Z^z = 1^ Z8z + (k), = 1^ Z{z -z) + (X-)2, (14) 



and 



I ZS'z = l Z{z-z) + 1 iZ{l-z) + Z(z-'z)}Px + {k)o. (15) 



Hence, subtracting, 



rZ8"z = I iZ(z - s) + ^(i - *s)) Px + {Tc)^. 



It is evident that if ^represent any fluxion except yr°i~-'' or ^o'-''i''^\ 

 there is no term on the right-hand side of order higher than {Icz), but 

 that if z be either of these fluxions, (s - z) is of unrestricted magni- 

 tude, and the term involving it becomes important. Therefore, the 

 only important terms at the -5-limit which arise from fluxions of an 

 order lower than S'^/^i^ or 8"y2'-°2^> axe 



which are now reduced to 



/ 



(ir«^(yi(°i) - yi(«i-0 + 2F„,^(y3(°2) - y.^^z^)) Px ... (16) 



