102 University of California Puhlications in Zoology [yoh. 19 



doubling of the units on the abscissa results in the curve being one- 

 half as long on the abscissa, while the height of the mode is far more 

 increased; furthermore, it serves to straighten out the descent of the 

 curve, giving a more typical frequency graph. 



This graph presents striking proof of the existence of a cycle of 

 encystment in Giardia in the rat, for we see that the average interval 

 between the modes of the cycles of encystment is about seven days. 

 Most of the intervals were about six days in length. In other words, 

 the maximum number of cysts were found in the faeces about every 

 seven days. This curve indicates that encystment falls into regular 

 periods, the climax of each recurring period being reached about every 

 seventh day. 



If a series of curves be so placed one above the other that the 

 first mode of each curve lies on the same ordinate line and the rest 

 of the curve be allowed to fall as it will, if there is a similarity or a 

 close identity of the interval between the other modes of each curve, 

 the modes of all the curves should, in the majority of the cases, fall 

 along as many common ordinate lines as there are modes. Since most 

 of the curves have three or four modes we should be able by such a 

 handling of the graphs as that given above to detect four common 

 ordinate lines upon which the majority of all the modes of the curves 

 will be located. The distance between these lines would be the true 

 interval between the crest of each mode, or the interval between the 

 days when the maximum numbers of cysts were found in the faeces. 



Such a treatment of the curves is the most conclusive proof of the 

 presence of similar intervals common for all the curves. Obviously 

 a high degree of exactness is impossible with the curves made from a 

 study of the cycle of encystment in rats, since the incidence of in- 

 fection varies in the case of each rat and there is also a margin of 

 error in the detection of the infection. However, there is, as we have 

 seen, evidence for an interval between the ejection of the maximum 

 numbers of cysts, and this interval can be seen by superimposing the 

 curves one above the other with their first modes coinciding, in order 

 to determine whether or not the other modes in the curves will in 

 the majority of cases also coincide on other common ordinates. 



In figure 16 the curves have been placed one above the other 

 so that their first modes lie on the ordinate line at point 6 on the 

 abscissa. In this figure of superimposed curves it will be seen that 

 the second mode for the majority of the curves is on an ordinate line, 

 at a place between points 11 and 12 or 13 on the abscissa; the third 



