Mr. H. A. Goodwin on a Property of the Parabola. 219 



overturn it." But supposing that my criticisms upon " the 

 published volume" could be set aside by the Professor's aban- 

 donment of his earliest researches, I think in having produced 

 a distinct public declaration of this fact it has done service to 

 science, and therefore to that extent my desire has been ac- 

 complished, for it cannot be said that there is in the published 

 volume any statement to the effect that those researches were 

 to be considered as superseded by the book ; so far otherwise 

 indeed, that we are told in the introduction that it is sent forth 

 " partly as a resume of previous researches which have from 

 time to time appeared, and partly as supplying what was 

 wanting to complete them," and more than once the early re- 

 searches are referred to in terms of approval. It is clear 

 therefore that without a distinct declaration, such as my letter 

 has drawn forth, neither I nor any other person would have 

 been justified in treating as discarded the researches in which 

 the author has stated it to be his opinion that " the refractive 

 indices are related to the lengths of waves, as nearly as pos- 

 sible according to the formula deduced from M. Cauchy's 

 theory." 

 August 11, 1842. 



XXXIX. Proof of Professor Wallace's Property of the Pa~ 

 rabola. By Henry Albert Goodwin, Esq.* 



To the Editors- of the Philosophical Magazine and Journal. 

 Gentlemen, 

 TF the accompanying proof of Professor Wallace's property 

 of the parabola appears to you to have any advantage over 

 former solutions in symmetry and conciseness, it is much at 

 your service. My object in offering it is to exemplify the 

 great use of the simple equation to the tangent, which I have 

 used, and because the method employed brings out the result 

 in a most direct manner. 



I am, Gentlemen, yours obediently, 

 Corpus Christi College, Henry Albert Goodwin. 

 Cambridge. 



Let aj a 2 « 3 be the tangents of the As which the three tan- 

 gents make with the axis of x. The equations to these tan- 

 gents are 



^ = «i *+—(!•) 3/ = " 2 *+ — (2-) </ = V+— (3.) 



a l ■ .■■■ a 3 



(1.) and (2.) intersect, .*. if x 1 y l be the coordinates of point 



r • . m a i+a9 



oi intersection #, = y, = m — — — : 



«j « 2 * a x a 2 



* On the subject of this paper, see p. 191 of the present Number. — Edit. 



