Mr. W. R. Birt on Storm-paths. 379 



=— j- II I — sinMsinw -{•■rsinucosv+T-cosujdRsmududv 



u€y rtfa* • , b* . c 2 \B . , , 



= —j L II ( — sinwsinu + -r-sinwcosu + 7-cos?< J-^smududv. 



Taking account of such terms only as give value in the final 

 result, this reduces to 



% sinududv '„ PPsmududv n /* s'mudu 



/*/*sinududv _ Pfsmududv _ /* sin udu 

 J J A " J J p + qcos2v~ ZJ \/pi—.q* 



a / 2 r 2 sm udu 



= 2 a be 1 7T / — ■ ■ 



J o \/(c 2 sin 2 u + 6 2 cos 2 u) [c 2 sin 2 « + a 2 cos 2 w) 



, P dx 



= abet: I — ■ 



J \/(a*+x)(b* + x)(c*+x) 



Multiply this by the factor -**•» which for convenience has 

 r * w abc 



/~*d x 

 been left out of the last steps, and we have vraSy / — for 



the result. By means of this result, (4.) becomes immedi- 

 ately 



v = „« y /"{i--£ * * )£. . (70 



J L a z + x b z + x c z + xj A v ' 



It seems not a little strange that this result should not by 

 the direct method have come out at once. I might give many 

 more examples in application of the theory given at the be- 

 ginning of this paper; but the above may suffice, the subject 

 being well understood. 



Gunthwaite Hall, January 27, 1846. 



LXI. On the Storm-Paths of the Eastern Portion of the North 

 American Continent. By William Radcliff Birt. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



IN the opening articles of the first and second numbers of 

 the second series of the American Journal of Science, 

 Mr. Redfield has contributed some further important informa- 

 tion to our present stock of knowledge respecting the storm- 

 paths of the eastern portion of the North American continent 

 and its adjacent seas. The chart No. 1, illustrating the arti- 

 cles, exhibits the tracks of sixteen hurricanes ; eight of these 

 form an interesting group distinguished by this striking cha- 

 racteristic : the paths are semi-orbital, presenting in each case 



2D 2 



