Mr. Robison on the Improvement of Lighthouses, 221 



P, Q are real functions of p and q, and the latter of odd di- 

 mensions in Q ; therefore a real value of q corresponding to 

 any proposed real value of p may always be found so as to 

 make Q = 0, and then we have R = P, a quantity always real. 



By assigning tojt? real values indefinite in number, the real 

 values of R thus obtained must either pass through zero,. or 

 else some one result is nearer to zero than any other. 



In the latter case putting p + h for jo, and supposing such 

 a value assigned to q that Q = 0, we should obtain a result 



R = P + A/A... (1) 



in which the lowest power of h alone is included, since we 

 may suppose h very small. 



But if we put for p 9 p + h I cos — |- V — I sin — J- this 



substitution will lead evidently to the same result as if we 



put for p, p -f h cos — | and for q, q + h sin — , and a real value 



— : and for q, a + h sin — 

 n * * n 



may therefore be found for q + h sin — , and consequently also 

 for q such that Q shall still vanish, and we get 



R = P + A it ^ cos * + V~^l sin * V 



t= P - Ah (2) 



One of the real values of R marked (1), (2) must necessarily 

 be nearer to zero than P, contrary to supposition. .\ Hence 

 it follows that R passes through zero ; that is, the equation 

 / (jt) as always admits a root of the form p -f q V ^— 1 

 where p) and q are real quantities. 



Caius College, Jan. 11. R. MURPHY. 



XXXVI. Suggestion regarding the Improvement of Light- 

 houses. By John Robison, Esq. Sec. R.S. Ed. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



AS there* are some circumstances which make it probable 

 that a light of large volume, though of moderate intensity, 

 should be recognisable at a greater distance during foggy 

 weather, than one of small extent, though more brilliant, and 

 capable of penetrating further in a clear atmosphere, it appears 

 to be desirable that some person who luis the means of doing 



