APPENDIX I. 495 



of reference. Let a ly ... a 6 and /? 1} ... /3 6 be the co-ordinates of a and (3 with 

 respect to A, B, C, &c. 



Let A lt ... A 6 be the co-ordinates of A with respect to the six principal Screws 

 of Inertia and similarly let B l , ... B 6 be the co-ordinates of B and in like manner 

 for C, 1), E, &c. 



An impulsive wrench on a of intensity a &quot; will have for components a &quot;^ on 

 A ... and a &quot;a 6 on F. These components on A, ... F may each be resolved into six 

 component wrenches on the principal screws of inertia, viz. 



a &quot;a 1 A l + o!&quot;o.^E^ ... + a&quot; a 6 F 1 , 

 a &quot;^ A., + a &quot;a^B. 2 . . . + a &quot;a 6 F 2 , 



But these impulsive wrenches give rise to an instantaneous twist velocity 

 about a whence by 80, we have, if A be a common factor, and a, b, c the principal 

 radii of gyration 



i = a 1 A 1 + a^S 1 + a 3 C\ + a^D l + a 5 E 1 + a 6 ^ 15 

 ttj AZ + a 2 B 2 + a. 3 C. 2 + a 4 Z) 2 + a 5 E. 2 + a 6 F. 2 , 

 hb(3 3 = a,A 3 + a. 2 B 3 + a 3 C 3 + a 4 D 3 + a 5 E 3 + a s F 3 , 



+ hc(3 5 =0.^ + a. 2 B 5 + a 3 C 5 + a 4 D 5 + a,E 5 + a 6 F 5 , 

 - hc(3 6 = a,A 6 + a. 2 B 6 + a 3 C s + a 4 Z&amp;gt; 6 + a 5 ^ 6 + a 6 ^ 6 . 

 Thus the linear relations are established. 



NOTE VIII. 



Remarks on 268. 



It ought to have been mentioned that the relation between four points on a 

 sphere used in this article is a well known theorem, see Salmon, Geometry of Three 

 Dimensions, 56 and Casey s Spherical Trigonometry, 111. 



It is also worth while to add that ^123 is the function which on other grounds 

 has been called the sine of the solid angle formed by the straight lines 1, 2, 3 

 (Casey, Spherical Trigonometry, 28). The three formulse of this article have been 

 proved as they stand for sets of six co-reciprocal screws. Mr J. H. Grace has 

 however kindly pointed out to me (1898) that the second of the formula; would be 

 also true for any set of five co-reciprocal screws, and the third would be true for 

 any set of four co-reciprocal screws. We thus have for a set of four co-reciprocals 

 p l sin 2 (234) + Pa sin 2 (341) + p 3 sin 2 (412) + p, sin 2 (123) = 0, 



where sin 2 (234) is the square of the sine of the solid angle contained by the straight 

 lines 2, 3, 4. 



