132 Prof. A. M c Aulay on Spontaneous Generation 



II. Enunciation of Mathematical Theorems. 

 Th. 1. If Q(«, /3) is linear in a and /3 then 



Q(£#0=Q(f?,?), 



where is any vector Unity. 



N B. — A quaternion linity is a linear quaternion ifunction 

 of a quaternion. A vector linity is a linear vector function 

 of a vector. When linity is used without qualification in 

 this paper it is understood in the more restricted sense of 

 vector linity. The logical custom would no doubt be] the 

 converse. 



Th. 2. If <f> is an arbitrary vector linity and ty a , yjr b are 

 vector Unities, then if 



yjr a will necessarily be equal to ^ 6 . If the same relation 

 holds with an arbitrary self-conjugate linity it follows 

 that ^a=^. 



Th. 3. In operating on a function of v equation (13) above 

 holds, 



<J = Ta (13) 



This is proved by first noting that i/ = ^(T'T— 1), and then 

 showing that for z a function of v 



dz = - SdTf dzl = - Sc/T£Ta~£. 

 Th. 4. If the linity <j> is given by 



(£&>= — /3Sft)a— /3'Sa)a / — ... =— 2£Sft>a, 

 then QG;«) = 2Q(«,£). 



In particular 



Q(fcxO=Q(Vi^i), 



Q(?l, x?i> s 2 , x? 2 ) = Q(Vl» ^l» V2> %), 



and so on. Thus we may always replace a pair \J a , r\ a by 

 2a, %f a and conversely. 



Th. 5. If <j> is any quaternion linity the following are the 

 line-surface and surface-volume theorems of integration. 



^ = fiWV<GVi, (14) 



JJ^s-JE^Vi* (15) 



In (14) </S is an element of an arbitrary surface, and dp. 



