14 



the operations here indicated being thus sure to make no 

 change in the part j3i, which is in the direction of the axis of 

 rotation, but to cause the other part /32 to revolve round that 

 axis a through an angle z= a. Again, let the same line /3' 

 revolve round a new axis of rotation denoted by a new vector 

 unit a', through a new angle a', into a new position )3'' ; we 

 shall have, in like manner, 



If we should make, for abridgment 



a 

 a tan — ^ — y, 



the formula (i) for any single rotation might be thus written, 



/3' = (1 + y)-' /3 (1 + y). (i') 



And if we then made 



(3 = ix + jy = kz, /3' = jx 4- jy^ + kz^, y=.i\-{- jfi + kp, 



i. j, k, being the same three rectangular vectors, or imaginary units, as in 

 the formulje (a) (b) (c), but x, y, z, x, y, z', X, fi, v, being nine real or 

 scalar quantities, we should obtain the same general formula for the trans- 

 formation of rectangular coordinates (with the same geometrical meanings 

 of the coefficients X, fi, v,) as that which Mr. Cayley has deduced, with a 

 similar view, but by a different process, and has published, with other 

 " Results respecting Quaternions," in the Philosophical Magazine for Fe- 

 bruary, 1845. 



The present writer desires to return his sincere acknowledgments to 

 Mr. Cayley for the attention which he has given to the Papers on Qua- 

 ternions, published in the above-mentioned Magazine : and gladly recog- 

 nizes his priority, as respects the printing of the formula just now re- 

 ferred to. But while he conceives it to be very likely that Mr. Cayley, 

 who had previously published in the Cambridge Mathematical Journal some 

 elegant researches on the rotation of bodies, may have perceived, not only 

 independently, but at an earlier date than he did himself, the manner of 

 applying quaternions to represent such a rotation ; he yet hopes that he 

 may be allowed to mention, that a formula differing only slightly 4n its nota- 

 tion from the formula (i) of the present abstract, with the corollaries there 

 drawn respecting the composition of successive finite rotations, had been 

 exhibited to his friend and brother Professor, the Rev. Charles Graves, of 

 Trinity College, Dublm, in an early part of the month (October, 1844), 

 which preceded that communication to the Academy, of which an account is 

 given above. 



