36 REPORT — 1871. 



subject to the t (2i+l) equations 



AB+A'B'+A"B" = 

 AC + A' C' + A"C" = 

 BC4-B'C' + B"C" = 



gives, from any one canonic gi'oup, another indeterminately. To find the degree 

 of indeterminateness, let absolute magnitudes in canonical forms be ruled by the 

 conditions 



SWd'r=^iWcI. = .. = l, 



we have therefore 



A=+A'^+A"^+. . . = 1, 



B2 + B'2+B"^ + . . . = 1, 



as in the ordinary transformation of rectangular axes in three dimensions. These 

 2/+1 equations with the i (2i+l) previous, amount in all to (i'+l) (2>+l) equa- 

 tions of condition among (2i-\-iy coefficients, leaving t (2j+1) independent vari- 

 ables. 



The only canonical form hitherto generally recognized is that of Laplace ; con- 

 sisting* of 2i+l polar hanuonics, of -which 1 is zonal, 2 (« — 1) are tessaral, 

 and 2 are sectorial. In the discussion which followed Mr. Cliflbrd's paper on 

 this form, I remarked that it seemed to be a singular case of the general canonic ; 

 notably singular in this respect, that for any one of its constituents the nodal cone 

 consists of circular cones having a common axis and planes through this axis. 

 The nodal cone of anj- spherical harmonic of degree i is an algebraic surface of 

 degree 2t + l, and I proposed the question, can canonical forms not be found in 

 which the nodal cone of each constituent is not resolvable into circular cones and 

 planes ? This question is answered by the present commimication. 



[A diagram was roughly sketched on the Doard, to illustrate the nodal cone of a 

 harmonic difl'ering infinitely little from a tessaral harmonic ; which, with 2i others 

 differing infinitely little from the other 2i—3 tessaral, the two sectorial, and the 

 zonal, constituting the polar canonic, would constitute a generalized canonic] 



General Phxsics. 



Account of Experiments upon the Resistance of Air to the Motion of Vortex- 

 rings. By EoBEET Sta^vell Ball, A.M., Professor of Applied Mathe- 

 matics and Mechanism, Hoycd College of Science for Ireland. 



The experiments, of which the following is an abstract, were canied out with 

 the aid of a grant from the Royal Irish Academy. A paper containing the results 

 has been laid before the Academy. 



The author proposed to bring this subject before the Association in order to elicit 

 discussion. He would greatly value am- suggestions as to the direction in which 

 future experimental researches would be likely to prove fruitful. Such suggestions, 

 though acceptable from all som-ces, would come with peculiar usefulness from 

 those who are conversant with the profound hydi-odj-namical problems of vortex 

 motion. 



A brief accoimt of one series of the experiments, and a Table embodying them, 

 will be given. 



Air-rings, 9 inches in diameter, were projected from a cubical box, each edge of 

 which is 2 feet. The use of this box was suggested by Professor Tait (see a 



* Thomson and Tail's ' Ivatural Philosophy,' § 781. 



