68 Cambridge Philosophical Society, 



primary division of the common median canal : this soon joins the 

 anterior division, and the common canal terminates at the median 

 opening below. Membranous tubes are continued from the three 

 osseous ones, and converge to terminate finally in the single Eusta- 

 chian orifice on the soft palate behind the posterior nostril. The 

 mucous membrane of the palate lines the various osseous canals 

 above described, and is continued by them into the lining membrane 

 of the tympanum." 



With regard to the homologies of the above described air-passages, 

 the author states that the lateral canals answer to the simple Eusta- 

 chian tubes of Lizards and Mammals, and that the median canal, 

 with its dichotomous divisions, is a speciality peculiar to the Croco- 

 dilian reptiles. 



The memoir was illustrated by nine drawings of the size of nature. 



CAMRRIDGE PHILOSOPHICAL SOCIETY. 



[Continued from vol. xxxvi. p. 240.] 

 March 11, 1850. — On the Numerical Calculation of a class of 

 Definite Integrals and Infinite Series. By Professor Stokes. 



In a paper " On the Intensity of Light in the neighbourhood of a 

 Caustic," printed in the sixth volume of the Cambridge Philosophical 

 Transactions, Mr. Airy, the Astronomer Royal, has been led to con- 

 sider the integral 



/»QO 



W 



/*°° TV 



== / cos — (w s — miv)dw, 

 Jo 2 



and has tabulated it from m= — 4 to m= -f-4 by the method of qua- 

 dratures. In a supplement to the same paper, printed in the fifth 

 part of the eighth volume, Mr. Airy has extended the table as far as 

 m= + 5'6, by means of a series proceeding according to ascending 

 powers of m. This series, though convergent for all values of m, 

 however great, is extremely inconvenient for numerical calculation 

 when m is large, and moreover gives no information as to the law of 

 the progress of the function for large values of m. The author has 

 obtained the following expression for the calculation of W for large, 

 or even moderately large, positive values of m : 



where 



W=2(3to)-*JRcos /V--Ws sin (<?--) j, 



R-l- 1-5.7.11 1.5.7.11 .13.17.19.23 _ 



1.2(72^5)2 + 1.2.3.4(72fl>) 4 



a _ 1.5 1.5.7.11.13.17 



l.lty 1.2.3(72^)3 



When m is negative, and -f» is written for — mw in the integral 

 W, so that in the altered form of the integral m is positive, there 

 results 



v ' I l.72p 1.2(729)2 J 



