~ 486 PROFESSOR C. NIVEN ON THE 
ments. of mathematics: in most cases the results run parallel.in the two groups. 
We may denote force-groups in general by the symbols (a8 y), (wu, v, w), 
&c., while stress-groups will be expressed by (A...F), (a... f), &c. In- 
variant functions will be represented by ume characteristic HS J, C05 * 
Theorem (e). 
ex 
Force-Groups. >... Stress-Groups « ‘ 
Lines (@, v, 2) (27 12 2 a YZ RE 
Since J.2°4+ Jy?+ J.2= Jr’ we 
have the akon On J, J, 0 0 ‘OY, 
i Tae . (7) 7772200 0) a3 
Moments of a particle with regard | hd 
to three planes. Nee di ae ah lee hme <2) 
Moments of a solid with regard Moments and negative’ eae 
to three planes. é of inertia of a solid. 
Forces. Stresses. 
Displacements. Strains and quasi-strains. | 
Theorem (f). Since di=5 d+ dy +" a ode, and dJ* = = By eo: 
ae = dF, it follows - that. (+ Ane 5 : =) J form a force-group and 
a aes ) J’ a stress-group. Moreover, as these characteristic 
: ad 1d : Paoe 
laws of resolution of 7. . . 5 7 depend on the relative directions only 
of the new and old fixed axes, and not on the subject of operation, 
we may remove the restriction that the latter is to be an invariant, and 
extend them to the case where it may be any directed eel In this 
dv dw 
way we establish the invariancy of such expressions pee = += a + ae 
a? ane a GMB a7F 
dx + dij? + ga, and 7 da + dy? +... + 2 Gray da dy * 
Theorem (g). If we choose J’= — J, we reproduce the group (A...F), 
while by choosing J’ = J, we see that (BC — D’,...., DE—CF) form 
a stress-group. This result enables us also to deduce the new 
invariants 
Tie VOB De) ee ee + 2f (DE — CE) 
Sigs bed?) (BO—D?) +)... + 2(de— 9) MECH), 

