its Analogues j and the Calculus of Imaginaries. 281 



of the problems, whose laws are represented, respectively, by the 

 equations (I.), (II.), (III.), it has been thought better to state 

 the solutions in their present form. It will be noticed that the 

 imaginary unit in the solution of (II.) has the same reference to 

 an unit circle, as that in the solution of (III.) has to an unit sphere. 

 Again, it is to be observed that we are not at liberty to write 

 the second form of the solution of (II.) in the shape 



r%Ae m i x+m & (cos m x z + i . sin m,z) (cos m 2 z -\-j . sin m^z) 



v =1 



^£Be TO i* +W22/ (cos m x z—i . sin m^fcos m^^j . sin m^s) f 



as at first sight it might be supposed ; and a similar remark will 

 apply to the second forms of the general solution of (III.). In 

 fact, neither in the Calculus of Triplets nor in that of Quaternions 

 does the ordinary property of exponential functions, namely, 



/(T)./(T')=/(T+T'), 



M) .f(Q!)=f(Q+Q!), 



hold good, unless T and T', Q and Q', be respectively co-direc- 

 tional*. In general, it will be seen that the statement of the 

 solution in the above form is equivalent to the breaking up of 

 the imaginary unit into its constituents, and to the illegitimacy 

 of this process the fact now mentioned reduces itself. 



5. As additional examples of the application of these imaginary 

 symbols to integration, it is proposed to show that by their use 

 we can obtain, very simply, integrals of several well-known dif- 

 ferential equations, and that they may be employed with advan- 

 tage generally when the equations to be solved are symmetrical. 



(I.) As a first example, let the solution of the familiar equation 



dx 2 + dy 2 + dz <2 = ds 2 , 



which has recently occupied so much attention with French 

 geometers, be sought. 



It is evidently equivalent to 



(dz + idx -hjdy) (dz—id% —jdy) = ds 2 , 



and the solution is given by the system 



z + ix+jy^fy^ + iy^) 

 z—ix—jy=yfr(s + ry 2 ) 



where y x and y 2 are arbitrary vectors. 



(II.) The general solution of the equation 



* Proceedings of the Royal Irish Academy, p. 433. 



