•33G Dr James Macaiiiay oti the Physical Geography, 



the two points A and B, it ought to have presented an appear- 

 ance of this kind. 



Let us retrace the steps of the successive improvements at 

 which I have hinted. First we have the function p=- (con- 

 stant), of which the first derivative is zero. Second, g=w/, 

 of which the second derivative is zero ; and, thirdly, g = w/ 

 (L— /) of which the third derivative is zero ; the softness of 

 the action increasing with the number of derivatives. For 

 further improvement, we must take a function of an order still 

 higher. Now, supposing these functions to be completely re- 

 soluble into factors, the curve given by them will have as many 

 points of reflexure (or of straightness) as the function has di- 

 mensions, and the curve will present a waved appearance. I 

 need not indicate to those who are acquainted with the higher 

 analysis, how the thorough investigation of such curves sur- 

 passes the present powers of that powerful science. Such per- 

 sons will at once recognise in the curve of sines, a transcen- 

 dental function, having an interminate succession of derivatives, 

 and whose form, easily submitted to calculation,] embraces all 

 the essentials of which we are in search. 



I therefore submit, as the proper sweep for joining two 

 straight parts of a railway, that portion of the curve of sines 

 which is contained on one side of its axis, and I subjoin a few 

 practical rules for its adaptation. 



Notes on the Physical Geography^ Geology, and Climate of the 

 Island of Madeira. By James Macaulay, A.M. and M. D. 

 Communicated by the Autho r. 



Few foreign stations are better known in this country, by 

 name and general description, than the Island of Madeira. 

 Its fruits and wines, and other productions of the soil, have 

 long enjoyed a merited celebrity. The remarkable salubrity 

 of its climate has been established by the number of invalids 

 who have there sought and found relief from disease, and pro- 

 longation of life. The boldness of its mountain outline, and 



