TRANSACTIONS OF THK SECTIONS. 4<8 



o = F (^(p.x,.. x„ (f (j?,). • • <?' (^n)) ; (12) 



it must also satisfy the following initial condition, 



= lim. f {a, a,, . . a„, <p — a, x^ — a,, . . x„ — a„). (13) 



s = B 

 Such are the most essential principles of the new method in analysis 

 which Sir William Hamilton has proposed to designate by the name of 

 the Method of Principal Relations, and of which, perhaps, the simplest 

 type is the formula 



S d s S s /I j\ 



i ax ox 



to be interpreted like the equations (8). 



The simplest example which can be given, to illustrate the meaning 

 and application of these principles, is, perhaps, that in which the dif- 

 ferential equations are 



and 



ddXj ddXq fny 



-j -; . K'^) 



axi dx^ 

 Here, ordinary integration gives 



!, = «, + a\ (s — a), Xj = «2 + a\ (s — a) ; (3)' 



and consequently conducts to the follovdng relation, (in this case the 

 principal one,) 



= (x, - «,)* + (x, - a,y -(s- ay. (4)' 



or 



s = a+ -/ (z, - a,)* + (^2 - aJS (7)' 



because, by (1)', we have 



a',« + «V = 1 ; 

 it enables us therefore to verify the relations (8), or (14), for it gives 



S s X, — a, d X, S d s 



Sx, s — a d$ Sdxy 

 and, in like manner, 



S s $ d s 



Sxi S dxj 

 Reciprocally, in this example, the following known relation, deduced 

 from (1)'. 



\SdxJ \SdxJ 



(10)' 



would have given, by the principles of the new method, this partial dif- 

 ferential equation of the first order, 



