138 Cambridge Philosophical Society, 



under the control of a law to which in reality they owe no obedi- 

 ence. 



An error somewhat analogous to this is shown to be committed 

 in the treatment of certain definite integrals, which are here submit- 

 ted to examination and correction, and some disputed and hitherto 

 unsettled points in their theory fully considered. The author is thus 

 led to what he considers an interesting fact in analysis ; viz. that the 

 differentials of certain forms require indeterminate corrections, in a 

 manner similar to that by which determinate corrections are intro- 

 duced into integrals ; and he attributes to the neglect of these the 

 many erroneous summations assigned to certain trigonometrical 

 series. This is illustrated by a reference to the processes of Poisson. 



The paper concludes with some observations on what has been 

 called discontinuity ; a term which the author thinks is sometimes 

 injudiciously employed in analysis, and prefers to treat discontinuous 

 functions as implying distinct continuities ; and by considering these 

 in accordance with the principles established in the former part of 

 the paper, he arrives at results for definite integrals of the form 



/+» 

 x~P dx totally diflferent from those obtained by Poiseon. Two 

 ■m 



notes are appended to the paper ; one explaining what the author 

 denominates insensible convergency and insensible divergency, and the 

 other discussing some conclusions of Abel in reference to certain 

 trigonometrical developments. 



March 1, 1847. — On the' Theory of Oscillatory Waves. By G. 

 G. Stokes, M.A., Fellow of Pembroke College. 



The waves which form the subject of this paper are characterized 

 by the property of being propagated with a constant velocity, and 

 without degradation, or change of form of any kind. The principal 

 object of the jiaper is to investigate the form of these waves, and 

 their velocity of propagation, to a second approximation ; the height 

 of the waves being supposed small, but finite. It is shown that the 

 elevated and depressed portions of the fluid are not similar, as is the 

 case to a first approximation ; but the hollows are broad and shallow, 

 the elevations comparatively narrow and high. The velocity of pro- 

 pagation is the same as to a first ajiproximation, and is therefore 

 independent of the height of the waves. It is remarkable that the for- 

 ward motion of the particles near the surface is not exactly compen- 

 sated by their backward motion, as is the case to a first approxima- 

 tion ; so that the fluid near the surface, in addition to its motion of 

 oscillation, is flowing with a small velocity in the direction in which 

 the waves are propagated ; and this velocity admits of expression in 

 terms of the length and height of the waves. The knowledge of 

 this circumstance may be of some use in leading to a more correct 

 estimate of the allowance to be made for leeway in the case of a ship 

 at sea. The author has proceeded to a third approximation in the 

 case in which the depth of the fluid is very great, and finds that the 

 velocity of propagation is increased by a small quantity, which bears 

 to the whole a ratio depending on the square of the ratio of the 

 height of the waves to their length. 



