264 Prof. Forbes on the Ascent of Mountains. 



height, as they respectively exhaust muscular power, that 

 empirical formula ought to be such a function of a, the angle 

 of inclination, that, 1, When a = 0, the distance shall be 

 four miles nearly, and the height ascended in an hour = 0; 

 2, When a = 90°, the height ascended, or distance x sine a, 

 shall be = 1000 feet neatly; 3, When « is between 20° and 

 30°, the ascensional effort or dist. xsin a shall be a maximum, 

 and = 1500 feet nearly; 4, That it shall vary slowly be- 

 tween 10° and 30°, and that below 10° it shall adapt itself 

 nearly to the experiments. 

 Now, a formula of this kind. 



= \ -' — ; 7^ — 6 sin a y sin a, 



\sm(a + fl) j ' 



sufficiently well satisfies these conditions ; h being the vertical 

 height in English feet attained in an hour, a and d being con- 

 stants, and 6 a constant angle. It will at once be seen that 

 the part within brackets measure.^ the distance traversed, and 

 determines the very rapid rate of diminution of the expression 

 for the first degrees ; a fixes the horizontal distance passed 

 over in an hour, and b the diminution of ascensional power 

 when the inclination is nearly vertical. Should this diminu- 

 tion be much smaller than we have supposed, the second term 

 might be independent of a, and should the maximum effect 

 be at a vertical inclination (which however is most impro- 

 bable) it would disappear altogether. The following numbers 

 seem best to satisfy my experiments : 



h = < - — 7 r^rr — 800 sin a > sin a in English feet. 



The following table exhibits the results of this law: 



A" (t1 f a t Space described Height ascended 

 ng seen . -^ ^^ hour. in an hour. 



feet. 



the result it will be seen is greatly above the mean effect. " 1832. Oct 15. 

 Left Weggis 2^ 5"". Angles of ascent i)^ . 15°, 15°, 8°, 1 1°, 16°, 1 1°, 15°, 

 2°, 12°. Mean ll°-4 Rigi— Culm. 4^ .35"'" The height is 4400 English 

 or at the rate of about 1800 feet per hour. 



