62 RECORD AND DISCUSSION 



Method of Reduction. — The method of reduction used is the same as that em- 

 ployed in the discussion of Kane's observations — it is by Lambert's improved 

 formula, so as to include the velocity of the wind, and not the relative frequency 

 alone. It is given in its outline in the article "Meteorology," in the 8th edition 

 of the Encyclopaedia Britannica. 



Let X &•> 6 3 be the angles which the directions of the wind make with 



the meridian (true), reckoned round the horizon, according to astronomical usage, 

 from the south, westward to 360°, a direction corresponding to that of the rotation 

 of the winds in the northern hemisphere ; and v 1 v^v. 3 its respective veloci- 

 ties, which may be supposed expressed in miles per hour ; and let the observations 

 be made at equal intervals (for instance, hourly). Adding up all velocity-numbers 

 referring to the same wind during a given period (say one month), and representing 



these quantities by v s 3 , the number of miles of air transferred bodily 



over the place of observation by winds from the southward is expressed by the 

 formula 



R s = S t COS 6i + s 2 COS 2 + 8 3 cos ®3 + 



And for winds from the westward 



R w = s 1 sin 0, + s 2 sin 6 2 + s 3 sin 3 + 



The resulting quantity 7?, and the angle <p it forms with the meridian, is found by 

 the expressions 



R = >/R? + R w ~, and tan <p = ,, 



The general formulae, in the case of eight principal directions 0, assume the 

 following convenient form : — 



R g = {S—N) + {SW—NE) Vl— {NW—SE) Vl 

 R w = (W—E) + {S W—NE) Vh + {NW—SE) y/1 

 "Where the letters S, S W, W, etc., represent the sum of all velocities during the 

 given period, or the quantity of air moved in the directions S, S W, W, etc., 

 respectively; l' s represents the total quantity of air transported to the northward, 

 and R K the same transferred to the eastward. These formulae, for practical working, 

 may be put in the following shape : — 



Put S—N =a S W—NE = c 



W— E=b NW—SE=d 



Then 



R a = R cos $ — a + 0.707 (c— d) 



R w = R sin ^> — h-\- 0.707 {c+d). 



Since R s , R,„, R, represents the quantity of air passed over during the given 



period in the direction 0°, 90°, Q°, respectively, we must, in order to find the mean 



velocity for any resulting direction, divide by it, or by the number of observations 



during that period; we then have 



y* = —, V w = — -, and V= — . 



A particle of air which has left the place of observation at the commencement 

 of the period — of a day, for instance — will be found at its close in a direction 180 

 -f- 4), and at a distance of R miles, equal to a movement with an average velocity of 



