Rev. J. Challis on the Brachystochronous Course of a Ship. 33 



24£ f ct. salt. 



132 measures -4 14I " 



133 



* — h 



{is 7 

 { 



{1 



ria§ 



\14| "~125 



S 26 



138 



£ 



147 27 



135 J 22 2 



gS * \23 



The diminution of the quantity of salt in the spring down 

 to a certain depth is a phenomenon even more difficult of 

 explanation than its subsequent increase below this point; and 

 it would seem very desirable to have similar observations 

 made in the case of other salt springs, with a view of ascer- 

 taining whether the same law holds good generally ; the total 

 absence of saline impregnation at a certain depth seems, in- 

 deed, hardly credible in the present state of our knowledge 

 on the subject. In the hope, therefore, of directing the atten- 

 tion of men of science to the question, I have been induced 

 to communicate to you the above details ; and now beg leave 

 to subscribe myself, Gentlemen, yours, &c. 



Oxford, Dec. 16, 1833. Charles Daubeny. 



X. On the Brachystochronous Course of a Ship. By the Rev. 

 J. Challis, Fellow of the Cambridge Philosophical Society*. 



T ■"'HE following theoretical considerations respecting the 

 ■*• most advantageous course of a ship are offered to the at- 

 tention of mathematicians chiefly as presenting some peculi- 

 arities of analysis, the author not being aware that this one 

 of the class of brachystochronous problems has been at- 

 tempted in the manner here proposed. 



I suppose the wind to be unfavourable, or the ship to be 

 obliged to tack, as the contrary case needs no inquiry. The 

 wind remaining of uniform intensity, we may assume the ve- 

 locity of the ship to be given, when the direction of its course 

 makes a given angle with the direction of the wind. 



Let O (fig. 1.) be a fixed point, OW a fixed line parallel 

 to the direction of the wind, which is supposed to blow from 

 W towards O. Then if the two straight lines POP', QOQ', 

 make supplementary angles, P'OW, Q'OW, with OW, the 

 wind will have equal effect whether the ship sail in the direction 

 POP' or QOQ'. If, therefore, O be the origin of rectangular 

 axes, OW the axis of x reckoned positively from O towards 

 W, and p = the tangent of the angle (0), which the direction 



• Communicated by the Author. 

 Third Series. Vol.4. No. 19. Jan. 1834. F 



