of the Linear Vector Function. 553 



within ellipsoidal geometry that have the same graphical 

 origin, and that have long been exploited. 



The derivation of (M) has led to an invariant form, that 

 makes (Ct 2 — Q<2 // ) necessarily perpendicular to the mean 

 vector (s), the second factor in the vector product being 

 their common value of (A). Otherwise the invariance is to 

 be delimited in the light of equations (14, 27). Therefore 

 every combination whose net sum is expressible by (M) must 

 at least be self-cancelling in the line of (s) ; the factor (A) 

 preserves for (M) a margin of flexibility proper to all vector 

 products. Observe how well all these features align with 

 the specialized equation (19). This mode of statement 

 adapts (M) to embody the dominant difference between a 

 pair of ''conjugate vectors," according to standard usage 

 of that term. For if we define a companion to (V) by 



V' = V-HL = Q, 1 + Q 2 ", .... (29) 



(Q 2 ) has been replaced with (Q 2 "). Then bring out further 

 the mutually supplementary relation of (V 7 , V) to (M) in the 

 symmetry of 



v-iM=a 1+ (<v'i-«) ; Y / +m=a 1 + {a 2 -iM) ; (30) 



and compare with the usual established forms. By reference 

 to equations (13, 20, 24), the nine coefficients in each set of 

 the rectangular components for (V, V) can be picked out, 

 and their interchanges recognized at sight, which render this 

 pair conjugate, (O^) being common to both. Proceed to 

 note that 



i(V + V')=a 1 + K^ + a 2 "); i(V-V') = p[; (31) 



showing complete formal resemblance to equations (9) 

 through the obvious identifications with (I/, A') in the 

 latter. The application is quite direct, if inquiry has 

 isolated an intrinsic axial constituent, as in electro- 

 magnetism : this treatment then acquires clear physical 

 status immediately. But though equations (20) are less 

 vital, if they merely prepare for deviations from equa- 

 tions (28) which simulate that experimental trend towards 

 composite vectors, the artifice does not lose its value. It 

 creates a widely comprehensive scheme *. What correlates 



* The range of the plan can be broadened useful])- to cover a 

 "Complex" of non-homogeneous vectors, like forces arid couples in 

 a fbrcive impressed upon any system as related to the centre of inertia. 

 Fusion into some analogue of screw-motion would follow easily. The 

 track pursued has several contacts with Poinsot's force-transfer, which 

 likewise turns upon shift of vectors to secure lever-arm. Our moment- 

 vectors (Q 2 , Q 2 ", M) drawn at (0) presuppose (A) "off centre." The 

 conditions underlying equations (34) "below are not altogether remote 

 from those of a couple. 



