112 On a Construction for the Range of Projectiles. 



and there is nothing to determine the direction of vibration at 

 the points coinciding with this tangent. We cannot therefore 

 make any assumption as to its constancy at different incidences j 

 there is nothing to show that it may not change from one inci- 

 dence to another ; and we know that it does undergo such a 

 change at the incidence of complete polarization. 



25. On the other hand, the formula (C/;') applies directly to 

 the phsenomcna without the aid of any subsidiary construction 

 whatever, unless indeed the whole principle of the reasoning hitherto 

 adopted be contested. 



But if we may still accept the reasoning of Fresnel and Arago, 

 of Airy and Lloyd, as valid, then it follow^s that both FresnePs 

 original formulas, and they alone, will apply directly to all the 

 experimental results without any extraneous considerations. But 

 one of them (A/) rests on an assumption as to the law of equivalent 

 vibrations different from the other (A') ; and the question remains, 

 whether that difference of assumption can be justiiicd, or whether 

 any other view of the theoretical principles can be found to lead 

 to the same results. 



I have thus stated in full detail the difficulties of this import- 

 ant case ; and will only add, that I shall look with great inter- 

 est to any attempts at removing them, which I hope this repre- 

 sentation may be the means of eUciting from those mathematicians 

 who have attended to the subject. 



XIV. Letter on Professor Galbraith's Construction for the Range 

 of Projectiles. By J. J. Sylvester, Professor of Mathematics 

 at the Royal Military Academy. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



PROFESSOR GALBRAITH'S geometrical construction for 

 finding the elevations of a projectile corresponding to any 

 given velocity and given range in a plane, horizontal or sloping, 

 is truly elegant, and, if new, constitutes a real acquisition to the 

 subject. It might be worth while for its accomplished author 

 to see if some analogous construction can be found extending to 

 the more general case where the field is a portion of a circle. I 

 need hardly add that the isoscelism referred to is, except for some 

 extreme suppositions (impossible to occur in practice), absolutely 

 independent of the form of the field. 



As well-con stnicted names are, in fact, condensed lessons, 

 lending an aid to the memory and imagination, of which modern 

 mathematicians are only beginning to appreciate the importance, 

 I suggest the following designations. 



