710 MR J. H. MACLAGAN-WEDDERBURN ON 



where \J/- is a completely self-conjugate function.* The integral is 



-if/p" = \dt<pa + 8 . 

 w J 



2. Homogeneous Equations — (Forsyth, § 16). — The general linear homogeneous 

 equation is 



<f>dp=f( P ,t) = t n ff(P 



• (2) 



where cpdp is a linear vector function of dp, whose constituents are homogeneous functions 

 ■of p and t of the n th degree and f(p , t) is also a homogeneous vector function of p and t 

 of the n th degree. 



Putting p = tv and expressing <p in terms of <r, we get 



<pa + t(p& = g((r) 



and consequently 



J 9(<r) - No- 

 where A is an arbitrary constant. JJa- is also arbitrary, making in all three arbitrary 

 -constants. 



3. Clairaut's Form — (Forsyth, § 20). 



P = t P +f(p) . . . (3) 



where f(p) is any vector function of p. 

 The solution is evidently 



p = ta +/(a) 



where a is an arbitrary vector. 

 An allied form is 



P = W) + 'Ap) • • ■ (3') 



Differentiating and putting y for ^ we have 



P=f(p)-SpV.(tf(p) + g( P )) 

 hence 



dt-t 



and therefore 



/ 



SdpV.f(p) 



SdpVJ(p) SdpV.g(p) 



M-p f(p)-p 



SdpV.f(p) „ [SdpV.f(p) 



f(p)-p 



t = Ae +e 



/ 



/(p) " P 



/- 



^)~P SdpV.g(p) 



J M-P 



4. Linear Form — (Forsyth, § 14. Tait, Proc. R.S.E., 1870). — The general linear 

 equation of the first rank may be written 



4>oP + <i>iP = >A a • • • ( 4 ) 



where (p <p-y and "^ are variable linear vector functions and a is a constant vector. For 

 the present we will only consider the case where (4) may be written 



p + < £<f 1< Pi p = <W a 



* Proc. B.S.E., xxiv., 1903, p. 410. 





