394 Prof. O'Brien on the Interpretation of 



can only reply that I defy Professor Challis to point out any 

 step of the reasoning by which the received equations are esta- 

 blished which requires the " preliminary step " above mentioned 

 to be previously taken, in other words, which would be invalid 

 did u, V, w mean the components of the whole velocity parallel 

 to the coordinate axes. With respect to the new axiom. Professor 

 Challis has of course a right to use his own words in his own 

 sense, and he may restrict it to the residual motion. But then 

 the enunciation is faulty, and it ought to be : — Let it be granted 

 that the directions of relative motion in each element of the fluid 

 mass may at all times be cut at right angles by a continuous 

 surface, the relative motion here spoken of being understood to 

 mean the resultant of the actual motion, and of a motion of 

 translation equal and opposite to that of the particle, which, at 

 the instant considered, is situated at that point of space which 

 has been chosen for origin of coordinates. If I have rightly in- 

 terpreted the somewhat vague statemeni; of Professor Challis, I 

 can only say that the axiom under discussion does not apply to 

 this motion. To take a simple instance of failure, letM= —mt/, 

 v = oax, 'w=f{r)—f{0), where »• = V' (a?- + ?/^) . It will be easily 

 found that in this case udx + vdij + wdz is not integrable by a 

 factor, except for two particular forms of the function /, a func- 

 tion which is not supposed to be restricted to any particular 

 form. 



Pembroke College, 

 March 11, 1851. 



L. On the Intei'preiation of the Product of a Line and a Force. 

 By the Rev. M. O'Brien, M.A., late Fellow of Caius College, 

 Cambridge, and Professor of Natural Philosophy and Astro- 

 nomy, Kinrfs College, London^. 



THE following interpretation of the meaning of a product 

 when one factor is a line and the other force, appears to 

 me to be one of considerable importance, and well deserving the 

 attention of mathematicians, for the following reasons. First, it 

 is an interpretation of great simplicity, based upon and imme- 

 diately deducible from the first principles of symbohcal algebra, 

 assuming nothing more, in fact, than the well-known generali- 

 zation of the sign + which makes AB + BC = AC, whether AB 

 and BC be lines drawn in the same direction or not. Secondly, 

 the results deducible from this interpretation are of great inter- 

 est in reference to geometry, mechanics, physical optics, astro- 

 nomy. Some of these results I gave in a paper read before the 



Communieated by the AutJior. 



