f ; [(27) 



552 Prof. F. Slate on a Graphical Synthesis 



other scale-factors applied to the same (# l9 y u s^) give 

 Q 2 " = (aJxjYi + (V*/i) v 2 + (c 2 ^i)v 3 = (%' + W + c 2 V i 



H= : Qj|--Qi , '==(a,--a s '}«iVi+(6 a -6 2 / )yiVs + (cj- 

 Thus the vanishing of (M) depends upon the equality in 

 pairs of tensors for the components of (Ct 2 , Q 2 ")- -^ so (Ml) 

 serves as a measured consequence of unequal scale-factoi s 

 there, on reduction to the same (ABC), through whatever 

 circumstances such differences come into question. 



The indications of reversal by equations (2) have prima 

 facie validity. Yet on proceeding to the algebraic equiva- 

 lents for ( — V) in equations (3), an issue of fact will never- 

 theless be raised : Whether phenomena harmonize with 

 making freely exchangeable, as related to the first forms, 

 the specific factorings in 



-V = a(-x i v i ) + b(- I/i vJ + c(-z 1 Y. 6 ), -i 



-V=(-aX^iVi) + (-&)<J/iV 2 ) + (-c)(* 1 y i ). J 

 Where physical properties sanction the first equation, they 

 legitimize at once unaltered scale-factors, to be partners of 

 what the reversed vector ( — 3s) stands for more widely. This 

 holds for coefficients that belong to a line, and not to one 

 direction in it ; moment of inertia and light-speed in 

 crystals are instances. The other equation refers more at 

 first-hand to opposed states like stretch and squeeze. The 

 proved precision of either equality Jeads to decisive infer- 

 ences in other directions. The consideration of such matters, 

 for which equations (13, 14, 25) may offer a starting-point, 

 bears upon classifying linear vector functions, especially as 

 regards conjugate pairs. The operand of the function that 

 here builds up (V) was indeed defined particularly by equa- 

 tion (21). However, with (a?j, y u z x ) of equations (3) 

 selected at will, and adopting the superpositions proved for 

 principal axes, the procedure enlarges itself at this stage 

 into treating functions of .any vector, whose components 

 (#1^1, y\ v 2? z \Yz) will then be chosen to represent. Primarily 

 by using for the operand some one scale-factor (planes 

 parallel to (ABC)), though the algebra may be shaped to 

 break down that restriction. The developments are adaptable 

 to whichever (3s) th^ working basis best yields, bearing in 

 mind about two differing scales, that either may be norm 

 for the other's distortion. It is true that equations (14) 

 are perhaps simpler for (n), another standard parameter of a 

 plane. Still the initiative at (s) or (n) will always be weighted 

 in favour of expressing physics ; a mathematical gain is less 

 fruitful. A thought that governs, too, the parallel sequences 



