176 On the Theory of Partial Differential Equations. [Nov. 24, 



V. " On the Direct Application of First Principles in the Theory 

 of Partial Differential Equations." By J. Larmor, M.A., 

 Fellow of St. John's College, Cambridge. Communicated 

 by Lord Rayleigh, D.C.L., Sec. R.S. Received November 

 8, 1887. 



(Abstract.) 



If an equation involving total differentials of any number of 

 variables can be expressed in the form — 



8u + (r hv = 0, 



where u, v are any functions of the variables, then the only single 

 integral algebraic relations that are consistent with it are included 

 under the form — 



u = 0(V). 



When the form of a is assigned, the functional symbol is to be 

 chosen, if possible, so as to agree with that form ; and if this is not 

 possible, then the equation has no integral expressible as a single rela- 

 tion. This statement holds because the equation expresses a particu- 

 lar case of the proposition tbat if Bu=0, then £v=0, and conversely, 

 i.e., that u remains constant (does not vary) when v is constant, and 

 only then, whatever be the particular values assigned to the variables : 

 but this is simply the definition of the algebraic idea of function- 

 ality. 



If, however, a involve differentials, the alternative 8u=0 when rr=0 

 may lead to integrals of a new type. 



In the same way, an equation of the form — 



Bu+a 1 ov + 2 Iw = 0, 



must have all its single integrals included under the form — 



u = 0(v, w), 



where the form of 0. is to be chosen so as to agree with the expres- 

 sions for <r l5 <r 2 , when these are assigned. 



When no single integral exists, equations of this type may be satis- 

 fied by two simultaneous integral relations, one of which may be 

 arbitrarily assumed, as originally pointed out by Monge. This kind 

 of exception, however, need not trouble us when partial differential 

 coefficients are concerned ; for these implicitly assume the existence of 

 a single relation connecting the dependent variable with the inde- 

 pendent ones. 



Traces of this idea are to be found throughout the writings of Boole 



