388 



Series] possessed the property that was exhibited as belonging to 

 the odd number 197. This was omitted in the copy from some 

 doubt whether it was universally true. Since the paper was read 

 that doubt has been removed, and it turns out that the property 

 belongs not only to all the terms of the series 1, 3, 7, 13, &c., but 

 to all odd numbers whatsoever. I am desirous to add to the paper 

 this statement by way of supplement. The property referred to 

 may be thus enunciated — 



Every odd number may be divided into square numbers (not ex- 

 ceeding 4) whose roots (positive or negative) will by their sum or 

 difference [in some form of the roots] give every odd number from 

 1 to the greatest sum of the roots, which (of course) must always be 

 an odd number. 



Or the theorem may be stated in a purely algebraical form, 

 thus : — If there be two equations 



a +b +c +c? =2r+l, 

 a, b, c, d being each integral or zero, n and r being positive, and r 

 a maximum ; then if any positive integer r' (not greater than r) be 

 assumed, it will always be possible to satisfy the pair of equations 

 w''--\-a^+y^ + z"=2n +1 

 w -\-x +y +z =2r' + l 

 by integral values (positive, negative or zero) of x, y, z. 



I hope shortly to communicate a proof of the above theorem, in- 

 dependent of any of the usual modes of proving that every odd 

 number is composed of (not exceeding) four square numbers. 



Note. — The differences of the roots of 197 were not fully stated 

 in the paper, I add them here : — 



197 



has 7 forms of roots : — 



o 



"14, 1, 0, 

 12, 7, 2, 

 12, 6, 4, 1 



3 ... 



7 







17. 



21 



o 

 u 



1. 3 ... 





9, 



11, 13 





19, 21, 23, 



''o < 



11, 6, 6, 2 



1, 3 







13, 





21, 





10, 9, 4, 

 10, 6, 6, 6 

 9, 8, 6, 4 



3, 5 









. 15, 



23 



a 



3, 5, 

 1, 3, 



7 







15, 17, 

 15, 





O 



7 





11. 



19 



January 19, 1854. 

 CHARLES WHEATSTONE, Esq., V.P., in the Chair. 



A paper was read, entitled " On the Geometrical Representation 

 of the Expansive Action of Heat, and the Theory of Thermo-d3''namic 

 Engines." By William John Macquorn Rankine, Civil Engineer, 

 F.R.S.S. Lond. and Edinb. &c. Received December 5, 1853. 



The author remarks, that if abscissae be measured from an origin 



