128 On representing by Diagrams the Results vf Observations. 



the numerical laws of the falling stars which appeared in the 

 various quarters of the heavens. The remarks, therefore, 

 which I have to offer apply equally to both those papers; and 

 indeed to all others in which the method of polar co-ordinates 

 is employed. In forming a diagram in the manner just de- 

 scribed, three objects seem to be proposed; to discover the 

 mean result, by drawing a curve line through the points A, 

 B, C, D, ...; from the inspection of such a curve to conjec- 

 ture the general laws of the phaenomenon ; and to be able to 

 interpolate results. These objects are gained in perfection 

 when rectangular co-ordinates are used, but not so when polar 

 co-ordinates are employed. Now when the experiments have 

 a relation to directions estimated about a point, and 'to mag- 

 nitudes depending upon those directions, polar co-ordinates 

 have a natural claim to be employed : but when used, as Mr^ 

 Knox and M. Gravier used them, there is a peculiar disad^, 

 vantage, which it is desirable to remove, especially as the re- 

 medy is extremely simple. In Mr. Knox's diagram, if a 

 radius-vector be drawn at random, it will represent the mean 

 quantity of rain which fell while the wind's direction was within 

 an angular distance of 22|° on each side of that radius-vector. 

 It is therefore impossible to see by inspection how much rain 

 fell while the wind's direction lay within two proposed direc- 

 tions, unless they happen to include exactly an angle of 45°. 

 It must be acknowledged that this is a great defect. The 

 remedy is as follows: — Instead of the lines, set off from the 

 fixed point, being taken proportional to the quantities to be 

 recorded, they should be taken proportional to the square 

 roots of those quantities. If this be done, the figure, drawn 

 through the points so obtained, has this property, — the area 

 included between any two radii-vectores represents to the eye 

 the whole result corresponding to all directions included be- 

 tween those two radii. For example; in Mr. 

 Knox's experiments, OA, OB, OC, OD ... , 

 being set off proportional to the square roots 

 of the corresponding quantities of rain ; and a 

 curve (as in the annexed diagram) being drawn 

 through the points A, B, C, D ... ; if we now 

 draw any two radii, OM, ON, making any angle with each 

 other, the area MON will represent the quantity of rain 

 which fell when the direction of the wind lay between M O 

 and NO. In M. Gravier's experiments, where number of 

 stars takes the place of inches of rain, the area MON repre- 

 sents the number of stars which fell in that portion of the 

 heavens whicli is inclmied between two planes, one of which 

 passes through MO and the observer's zenith, and the other 



