120 Mr. F. L. Hitchcock on the 



functions), provided such forms are what were just above 

 called ordinary functions. For example, 



d& = dq.es + dUVq.(±TVtP—Tyq.#), . . (16) 

 and similarly 



ds'm q = dq . cos q + d\JYq . ( + TV sin q — TYq . cos^), (17) 



and so on. If the plane of q is fixed, so that JJYq is constant, 

 it is evident that the right side of (15), or of any special case 

 of it, reduces to its first term. The presence of the term, or 

 terms, in dXJYq is due to the fact that the quaternion 

 expansion is valid for space of three dimensions, so that the 

 ordinary formulae have to be generalized. This is an illus- 

 tration of the truth referred to in Art. 3 that the Quaternion 

 Calculus is an extension of the system, due to Buee, Argand, 

 and others, by which a complex quantity is represented on a 

 plane. It is the only such extension which preserves both 

 the distributive and the associative laws. 



In (15), or in any special case of it, we may replace d 

 by V, thus, 



VQ = V?.DQ + VUV?.(±TVQ-TV?.DQ), . (18) 



Q being any ordinary algebraic or transcendental function of 

 the quaternion q. As an instance, 



V log q = \7q.q- l + VUV? . ( ± TV log q - TYq . q~ l ) ; (19) 



TV log q as defined by Hamilton is the angle, or the ampli- 

 tude, of q. 



More generally , if FQ be any quaternionic function of Q, 

 we have 



dFQ=fdQ, 



where / is the linear function obtained by differentiating Q. 

 Then by (15) 



dFQ =f[dq .DQ + dXJYq . (±TYQ-TY g .DQ)], (20) 

 and by (8) 



VFQ = Vf[q'BQ + VYq / .(±TYQ-TYq.J)Q)-]. (21) 



As an example under the last two results, we have, if a and 

 b are any two constant quaternions, 



d(a sin bq) = abdq . cos bq + adUYbq . ( + TV sin bq-TYbq . cos bq\ (22) 



obtained from (20) by putting a for / and bq for q. And 



by (21), 

 V'a sin bq' = Vabq' cos hq + V'aU Vbq'. ( ± TV sin bq - TVbq . cos bq) . (23) 



