21 



x. y. z, is understood the set of equations m 



a b c 



a'b'c' 



x y / x y z 



or, in full 



-b' x). This 



m 1) / — c y. c x a z, a y — b x)=(b' z — c' y, c' x— a' /., a' y- 

 corresponds to vector multiplication in quaternions. 



The analytical methods thus perfected are, in fact, a sort of degraded 

 and cumbersome quaternion notation in which (a, b, c) stand for 

 ai 1) j-fc k, etc. It involves the necessity of thinking by steps parallel 

 to the axes, and when results are obtained it involves the fitting together 

 of the various steps in order to see what is the actual state of affairs in 

 space. To do this requires considerable practice and grasp of technique, 

 all of which is avoided in quaternions. For example, equations (S) were 

 unnecessary in quaternions, the results desired being sufficiently evident 

 from (7): while even after (8) is derived the technique of equations of the 

 first degree must be at command before the results stated can be seen 

 in the analytical method. The letters m ls m_, in (9) and on are not the 

 masses of ' 1) . . . (5). 



eolations up motion. 



id'x, d" y, d 2 z x \ m, m v 

 (1) m. ! 3TT> xiri-xir ! p-"(x,y,z) m t ( 



dV, 



d t ' d t ' ' d f 2 j 



(2) m., C — J - ^- — Z = 1 - m ' m 



(x ; y, z) 



I d r - d t 2 d t" I 

 where (x, y, z)=(x 2 — x,, y, — y,, z, — z, ) where/ 3 



and r x y / •' >' 



Adding (1), (2), also dividing out common m's and subtract- 

 ing, putting M=mj -J- m, , we have : 



TV 

 d 2 /° 2 _-m 1 m. 



(3) ( m 



d 2 X! 



d 2 xr 2 

 dt 2 



1 ,,UU " mi ^ + m ^ 



I 



d'x d'y d 2 z 



d V ' d t J ' d t 2 



M 



(x, y.z) 



dt 2 



M 



EQt'ATIONS OK MOlIo.v INTEGRATED. 



Integrating ( 3) twice, we have : 

 (5) (m, x,- : -m, x,, ....... )== m, ,v m.,-. 



i at b, a' t—b', a" t b" I 



a t - ,-. 



Hence, the center of gravity moves in a straight line with uni- 

 form speed, viz: 



