DURING POST-NATAL DEVELOPMENT. 533 



" 



the cube of the height we get a figure which also expresses the ratio of "average 

 cross-section of the body to square of height, since the height in inches is a diameter 

 common to the two volumes compared. Thus, if v = volume of body, h = height, 

 R = ratio of volume to cube of height, r = ratio of cross-section to square of height, then 



v h 3 v 



R = v:h 3 = -r:-r = -r: h? = r 

 h h h 



The square root of this ratio gives what may be termed the ratio of "average 

 transverse diameter to height, thus:_ 



" 



In use of the metric system we get similar ratios if we calculate volume from weight 

 in grams and specific gravity at 1.02516. 



Instead of using the ratio between height and the diameter of an estimated 

 volume we may make direct use of the height-weight ratio or of the height-weight 

 index of build to obtain a mathematical expression of relative transverse diameter. 

 The ratio of weight in pounds to cube of height in inches has the same value as the 

 ratio of "average" cross-section of the body to square of height, as the following 



equation shows: 



W H 3 



W:H 3 = -fj- : -77- = height-weight ratio. 

 ti n 



The square root of this ratio expresses a ratio of transverse diameter to height. 

 Assuming that a pound equals 27 cubic inches, this ratio would be 



, - 



'\r i 



of the cubic-inch ratio described above. Thus, for the infant shown in figure 1 the 

 square root of the height-weight ratio 0.0009178 is 0.030296, while the ratio of the 

 square root of the cross-section in inches to the stature is 0.15742, which is 5.1962 

 times as great. 



The height-weight index we have used is equal to the height-weight ratio X 1000. 

 Since this is equivalent to the ratio between weight in pounds and a cube of the 

 tenth of the height in inches, it is necessary to divide this index by 10 or to multiply 

 the height-weight ratio by 100 in order to get an equivalent index to express the 

 relation between average cross-section and square of height. Index 0.918 thus 

 becomes 0.0918 if used to express the transverse section ratio. We may call this the 

 transverse section index. The square root of this index gives us a transverse diameter 

 index which has 10 times the value of the transverse-diameter height ratio described 

 above. Thus the ratio of the square root of the transverse section of volume esti- 

 mated in inches from weight in the infant shown in figure 1 is 0.15742. In terms of 

 the square root of the height-weight ratio it is 0.030296. The transverse-diameter- 

 height index is 0.30296. This last ig the more practical for ordinary use in dealing 

 with inch-pound units. It is equal to about twice the square root of the transverse 

 section estimated in inches and hence is equivalent to the long diameter of a cross- 

 section 4 times as broad as thick. A curve based on this diameter ratio, the square 

 root of the tenth part of the height-weight index, enables us to compare the curves 

 of the ratios of linear measurements to stature with one expressing the ratio of a 

 theoretical transverse diameter to stature. We have accordingly plotted such a 

 curve " X " in charts I and J. 



