On a Theorem in Plane Kinetic Trigonometry . 471 



take the trouble to look in the right places : for there cannot 

 be any ground for the doubts cast upon the subject in the 

 articles above quoted ; seeing that the ignis fatuus is produced 

 by the ignition either by lightning or by accidental flame of 

 light carburetted hydrogen or marsh-gas which is generated 

 by the decay of vegetable matter in boggy places ; and the 

 reason why the phenomenon is now so seldom seen is that these 

 places have been drained and brought under cultivation. 



Highgate, N., 4th August, 1891. 



LVI. On a Theorem in Plane Kinetic Trigonometry suggested 

 by Gauss's Theorem of Curvatura Integra. By Sir William 

 Thomson*. 



1. A LBERT GIRARD'S beautiful theorem of the " Sphe- 

 -£-L rical Excess/'' in spherical trigonometry published 

 about 1637, and used practically 150 years later by General 

 Roy in the trigonometrical survey of the British Isles, was 

 splendidly extended by Gauss in his theoremf of the " Curvatura 

 Integra." There must be a corresponding theorem in the 



" kinetic trigonometry " suggested in the marginals of 

 Thomson and Tait's ' Natural 'Philosophy/ §§ 361 (a) (b) (c) 

 (d), for the motion of the generalized conservative kinetic 

 system of any number of variables. For the very simple case 



* Communicated by the Author. 



t " Disquisitiones Generates circa Superficies Curvas; auctore Carolo 

 Frederieo Gauss ; Societati Regiae [Gottingensi] Oblatce d.viii. Octobr. 

 mdcccxxvii." Collected Works, Vol. IV. Gottingen 1873. Thomson 

 and Tait's Natural Philosophy §§ 131 ... . 138. 



