Royal Society. ^361 



of a class of simultaneous differential equations of the first order, 

 including as a particular case the form (which again includes the 

 dynamical equations), ona 



dyl •^' dxl ^ ' 



where o", ... j"„, y, ... y„ are two sets of n variables each, and accents 

 denote total differentiation with respect to the independent variable t ; 

 Z being any function of ^'| &c., y^ &c., which may also contain t ex- 

 plicitly. The part now laid before the Society is limited to the 

 consideration of the above form. 



After deducing from known properties of functional determinants 

 a general theorem to be used afterwards, the author establishes the 

 following propositions. 



If J-, ... x„ be w variables connected with n other variables y, ...w 



j-y 



by n equations of the form y(= —— (X being a given function of 



a;, ...arj; then the equations obtained by solving these algebrai- 

 cally, so as to express .Tj ... x^ in terms of y, ... y„, will also be of the 



form a;,= -— ; where Y is a function of y, ... y„, which may be de- 

 fined by the equation 



Y=-(X) + (x,)y,+ ... +(.i-„)y„, 

 in which the brackets indicate that the terms within them are to be 

 expressed as functions of y^ ...y„- Moreover, if p be any other 

 quantity contained explicitly in X (besides the variables a-, ... a:„), 

 the following relation will subsist ; namely, 



dX dY _ 

 — 4- — =0, 

 dp dp 



the differentiation in each case being performed only so far as^ 



ajjpears explicitly in the function. 



It is then shown that if X contain explicitly, besides x^ ... x„, the 



n constants a^, a^, ... a„, and the variable t, and if the 2n variables 



a?, ... x,, y, ... y„, be determined as functions of t by the system of 



2b equations, 



;^=^" d^,=^- <"•) 



where 6, ... i„ are n other constants, the elimination of the 2w con- 

 stants from these equations and their differentials with respect to t, 

 leads to the system of differential equations (I.), if for Z be put the 



result of substituting in — — - the values of the 2m constants in 



terms of the variables. The equations expressing the 2n constants 

 in terms of the variables may be considered as the 2n integrals of 

 the system (I.). 



The author employs the symbol [p, q'] in a sense similar to that 

 in which Poisson and others have employed (p, q), namely, as an 



abbreviation for i;,( Ji '^i- JLil.) ; and he shows that if p, a 



^dyi dXi dXi dy/ * 



