356 Mr. A. Cayley on a Theorem relativg 



The effect would not depend upon the quantity of salts con- 

 tained in the freezing water, though its degree would. The 

 proportion of salts necessary to be added to pure water to lower 

 its freezing-point 1° F. may be very sensible to chemical tests ; 

 but the proportion required to make the difference y^^th or 

 To'u 0^^^ of a degree woidd be far less : and if we suppose that 

 only vjo^h of a piece of ice is brought into the condition of melt- 

 ting before the rest of the mass, and that the salts in that pro- 

 portion were originally in the whole of the water, then its quan- 

 tity there may be so small as to escape detection except by very 

 careful analysis. However, it would be desirable to examine the 

 water chemically which is produced by ice distinguished by 

 having in its interior much that liquefies before the rest. 



It is easy to mate ice perfectly free from air, and, as I believe, 

 from salts, by a process I formerly described. It would be in- 

 teresting to see if such ice had within it portions melting at a 

 lower temperature than the general mass. I think it ought not. 



Ever truly yours, 



M. Faraday. 



Royal Institution, Dec. 9, 1857. 



XXXVIII. On a Theorem relating to Hijpergeometric Series. 

 Bij A. Cayley, Esq.^ 



N attempting to verify a formula of Hansen's relating to the 

 development of the disturbing function in the planetary 

 theory, I was led to a theorem in hypergeometric series : viz. 

 writing, as usual, 



■ni a \ 1 «'/3 , « . a+1 . /S./S + l „ 



^{^, /3, y, x) = l+ ^x+~^ — li ^ +•• • 



1.7 1 .2 .y .y+1 



then the product 



F(«, /3, 7 + 1, a;)¥{y-u, y-^, 7 + i, .v 

 is connected with 



(l_^,)-(v-«-^)F(2«^ 2/3, 2y, x) 



by a simple relation ; for if the last-mentioned expression is put 

 equal to 



then the product in question is equal to 



7 + ? 7 + -}-7 + t 7 + i -7 + 1 -7 + 1 



* Communicated by the Author. 



