PHYSICS: A. ST. JOHN 
193 
10 ( 1,3,5,... K j m=0 
2,4,6 i w=0 ) 
«? 3 = i + s 2 ~~t~ - 2 2J («* 9"*^ (12) 
(l,3,5,... » i w=0 
CO I 
+ 2 t2 2 +i) * ^ • 
2,4,6,. . . i m=0 ) 
The interpretation of j and /x is as before, except that, when k is even, j is a 
residue of one of the numbers \k, \k + l 2 , . . . , \h + (k — ' l) 2 . These 
identities embody the theory for 7 or 3 squares. It should be noted however, 
that the application of my method becomes very much more difficult when 
s = 3, as the double series used are then not absolutely convergent; and in fact 
the only proof of (12) which I possess consists in an identification of the results 
which it gives with those already known. 
I conclude by a word concerning the cases in which s>8. Here, when s is 
odd, we are on untrodden ground. We have the asymptotic formula (6); 
and we can evaluate X s (n) as when s = 5 or 7, thus obtaining a series of new 
results. But it is no longer to be expected that our results should be exact, 
and I have verified that, when s = 9, they are not exact, even when n = 1 . 
1 Glaisher, J. W. L., Proc. London Math. Soc, (Ser. 2), 5, 1907, (479-490). 
2 Ramanujan, S., Trans. Camb. Phil. Soc, 22, 1916, (159-184); Ibid., (in course of 
publication). 
s Mordell, L. J., Quart. J. Math., 48, 1917, (93-104). 
4 Hardy, G. H., and Ramanujan, S., Proc. London Math. Soc, (Ser. 2), 17, 1918, (in 
course of publication). 
THE CRYSTAL STRUCTURE OF ICE 
By Ancel St. John 
Department of Physics, Lake Forest College 
Communicated by R. A. Millikan, April 30, 1918 ' 
During the winter of 1916-1917 the crystal structure of ice was investi- 
gated by means of the X-rays. The photographic method originated by 
deBroglie 1 was used with certain modifications suggested privately by Dr. 
A. W. Hull. The source of energy was a Coolidge tube with tungsten target 
excited by an induction coil with mercury turbine interrupter. At first the 
