4S6 TRANSACTIONS OF SECTION A. 



this constant term lias the $a7ne value for all. The system E,., „ of lowest order (r ), 

 in which this constant term occurs, has usually a non-periodic portion of length {h') 

 found as above. In passing to higher orders (>• + x) this unique portion decreases 

 in length (h') with increase of .r, until at last it vanishes {h' = 0), and the residue- 

 system of E,.+.,., „ is reduced to the constant-term R,+.r,n (of same value as 

 already found, R,-, /,4i) ; and in all higher orders the residue-system consists of this 

 same constant-term alone. 



Hence, taking the column (E,., o) of zero degree of the E,., „ system, its residue- 

 system R,, consists of a non-periodic part (Ro.oj Ri.ot • • • Ra.oj where h = h' — \ 

 (the h' just found), and a constant term R/, + i, « = tie constant term just found. 



All moduli (m) which have the same ^i have the same period-lengths h\, Av in 

 their rth order Er, „ residue-systems. 



Thus all the specific factors of (2' — 1), and also any product thereof, have the 

 same |,, .and have the same train of |^,, »;,,, and also the same period-lengths A',, A",- 

 when used as moduli. 



4. The Different Kinds of Integrals of Partial Differential Equations. 

 By Professor A. R. Forsyth, Sc.D., F.R.S. 



The customary classification of Integrals of partial differential equations of the 

 first order is originally due to Lagrange. The different kinds of integrals to which 

 it leads are called general, complete, and singular respectively. In the case of a 

 linear equation in two independent variables, taken to be 



Vp + Qiq = V. ..,,,,(]) 



with the usual notation, the singular integral does not exist; and the general 

 integral is regarded as the most important, on account of a proposition which 

 attempts to prove that this general integral is completely comprehensive. To 

 obtain it, we denote by 



u - u (.r, !/, s) = constant, r = v (.c, y, s) = constant, 



two independent integrals of the equations 



dx _dy _dz ,-,, 



P~Q~R ^-^ 



The general integral is then given hjf(u, i') = ; and the proposition asserts tliat, 

 if \//- = be any integral of the partial equation, a form of / can be chosen such 

 that ^ =/(?', ^')- 



If this were vmiversally true, undoubtedly the general integral would be com- 

 pletely comprehensive ; but it is not true, for the usual proof of the proposition 

 depends upon a fallacy which was, I believe, first noted by Goursat and by 

 Chrystal. Briefly stated, this usual proof is as follows: Because m = constant and 

 V = constant are integrals of the equations (2), the relations 



P^ + Q^ +Rp, =0, 



ox 01/ OS 



o.v oy oz 



are satisfied identically; and because >/' = gives an integral of the equation (1), 

 the relation 



ox oy oz 

 is .satisfied. But this last relation is not necessarily satisfied identically ; the 



