234 Proceedings of the Royal Society of Edinburgh. [Sess. 
and no other independent relations than these connect the six quantities 
A' 3 , B' s , C 3 ', D'* E' 3 , and R ' 3 . 
We can therefore express the result in terms of A' 3 , C' 3 , and R' 3 , and 
thus get 
- 3150' + 30R'](21( a 4 + /3 4 , - 17) . . . (24) 
The dotted curve in fig. 5 represents (24) on an arbitrary scale. The 
relative sizes of its lobes are strongly contrasted with those of the similar 
curve characteristic of the cubic arrangement. 
The transverse component, of this order, is 
2-3 
M (a? 
sin 0 . P' . 
= l) • sin 0 cos 0[231(6 cos 4 (9) - 315(4 cos 2 0) + 105(2)](A 2 + p 2 //. 2 + gV) 7/2 . 
The A-component of this is (cf. § 13) 
A - a(aA +p/3/x 4- qyv) 
. 
(aA +pfifx + qyv) 
J 
” 1/2 
.^(aA + + qyvY 
33 
3 0 (aA + pPn + qyvy + 5 
/ 
/ 
- 7/2 
In the cubic system this reduces to 
M fa 
“ ' 42 {33[(a 4 (l - a 2 ) - 0 * - /)A' S - 15(a 4 (/3 2 + y 2 ) + /3 4 (|g + a 2 ) + y 4 (a 2 + ,8 2 ))B' 3 
+ 5(/3 4 + y 4 + 2a 2 (/3 2 4- y 2 ))B' 3 + 30(1 - 
- 30[(a 2 (l - a 2 ) - /3 4 - y 4 )C' 3 - 6(a 2 ,8 2 + /3 2 y 2 + y 2 a 2 )D' 3 + 3(/3 2 + y 2 )E»' 3 ]}. 
On limitation of the investigation to the plane y = 0 we have 
( = /3 2 a . 42“(^) 2 {33[(a 2 - /3 2 )A' S - 15a 2 B' 3 + 5/3 2 (l + a 2 )B' s ] - 30[(a 2 - /3 2 )C' S - 6a 2 D' s + 3D'„]}. 
By interchange of a and fi we obtain the yu-component q, and the resultant 
transverse component of the internal force, acting in the plane y = 0 in the 
direction from the X-axis to the yoi-axis, is r\a — %/3. By evaluation of this 
we get the result 
v T' 4 = 21^^) 2 <x/ 3(/3 2 - a 2 )(231A' s - 315C' S + 30R' S ) , . . (25) 
the prefix y referring to the condition y = 0. 
It is to be noted that the bracket involving A' 3 , etc., is identical with 
the corresponding bracket in (24). Although, by means of the above 
equations connecting A' 3 , B' 3 , etc., it is possible to assign various inequalities 
