December 1, 1894.] 



KNOWLEDGE. 



281 



of miscellaneous insects which only visit flowers occasionally 

 and at random, even if the latter be equally adapted as 

 regards bodily form. An insect in any case cannot eti'ect 

 cross-fertilization at all unless it visits at least two tlowers. 

 A plant will, therefore, lose rather than gain by attracting 

 any species of which the individuals do not, daring the 

 time the plant remains in bloom, visit on an average at 

 least two flowers apiece. The superiority of a small 

 number of industrious insects over a larger number of 

 indolent ones depends chiefly on the increased risk in the 

 latter case of repeated visits being paid to the same flower. 

 Ad individual bee of any intelligence will not, if it can help 

 it, return to a flower from which it has already removed 

 the nectar. If, however, there be a number of visitors, 

 manifestly they cannot avoid entering many previously 

 visited flowers, and in this way a considerable proportion 

 of the visits are as good as lost so far as the blossoms are 

 concerned. To illustrate the disadvantage arising from 

 revisitation or. overlapping, let us suppose that a certain 

 area contains ten flowers ; the visitation of the whole ten 

 will be overtaken in a given time by a bee which makes 

 ten visits in that time, assuming that the insect works 

 intelligently so that no flower receives more than one visit. 

 But ten miscellaneous insects, say flies, each visiting but 

 one flower in the same period and working independently 

 cannot, although the total number of visits is the same, 

 overtake the whole ten flowers. Obviously there is nothing 

 to prevent all the ten flies entering the same flower, and in 

 that case the other nine would remain unvisited. Again, 

 their visits might be confined to two, three, or more 

 flowers, while the remaining ones were neglected. The 

 ways in which the ten visits may be distributed are so 

 numerous that it could only rarely happen that each fly 

 went to a separate flower and the whole ten were visited. 

 The first fly runs no risk of entering a previously visited 

 flower ; the value of its visit may, therefore, be represented 

 as 1. The value of the second fly's visit is rather less, for 

 although there are still nine unvisited flowers, it may 

 chance to enter the same one as its predecessor, and the 

 visit be lost. As there are ten flowers, the risk of this 

 is y'o ; the value of the second fly's visit is, therefore, 

 diminished by that amount and only counts y%._ The 

 third fly is also liable to enter the first visited flower, and 

 there is the further contingency of its entering the same 

 one as the second fly ; the former risk, as before, is [\ ; 

 the latter ^'jj of i",,, or , ",g, which falls to be deducted along 

 with J^ from 1 to obtain the value of the third visit, which 

 represents the average efficiency of the third fly. We have 

 thus obtained the first terms of a decreasing geometrical 

 series !• +V'o+-i'5',>+ iWo) ^^'^■^ ^^^ successive terms of 

 which represent the value of each additional visitor. 



From the formula s= "'' ^'' , where s is the sum of the series, 



(( the first term, ;■ the ratio, and n the number of terms, 

 we can obtain the value of s — that is, the number of 

 flowers overtaken by n visitors. In like manner, insects 

 which each visit two flowers furnish a corresponding 

 series, 2, 1-6, 1-28, etc., and similarly with those having 

 a higher rate of industry. Arranging these results, we 

 get the following table, in which the first vertical column 

 shows the number of insects engaged, and the upper 

 horizontal line their rates of industry ; the other figures 

 indicate the average number of separate flowers overtaken 

 in each case. 



From this table it will be seen that while one flower is 

 visited in the specified time by an insect whose industry is 

 1, and two by an insect with twice this diligence, two 

 insects each with unit industry can only overtake on the 

 average 1-9. It will also be seen that ten casual visitors 



only overtake 6-5 out of the ten flowers in their ten visits ; 

 the diflerence 3'5 represents the average loss from over- 

 lapping of agency or frequenting previously visited flowers. 



1 23 45 6 789 10 



The efficiency of five insects whose industry is 2 is slightly 

 greater, being (i-7 ; the total number of visits is the same 

 in both cases, but more are overtaken in the latter, since 

 each insect must visit two distinct flowers. Again, when 

 two insects are engaged, each visiting five flowers in the 

 specified time, they overtake 7-5 out of the ten flowers in 

 their ten visits. Lastly, if we suppose only one insect to 

 be employed with an industry of 10, then all the flowers 

 are overtaken, assuming that it works intelligently and 

 visits each flower but once, for a careless or unintelligent 

 worker will lose as much as ten diflerent insects do by 

 overlapping. The above figures also show that eight 

 visitors with an industry of 2 overtake exactly the same 

 amount as five, each having an industry of 3 : there is an 

 extra loss of one visit from overlapping in the former case. 

 As the result of these calculations we deduce the following 

 conclusions : — 



1. The maximum eft'ect is obtained when one insect is 

 employed to visit the whole ten flowers. 



2. The industry of the individual visitors is of greater 

 importance than their numbers. 



3. When more than one insect is employed the efficiency 

 is rapidly reduced at first, afterwards more slowly ; this 

 means that there is a premium on the higher rates of 

 industry. 



4. The efl'ect of augmenting the industry diminishes as 

 the number of insects employed increases ; Avhere the 

 number of visits is greatly in excess of the number of 

 flowers, overlapping must occur whether the excessive 

 visitation be due to numbers or industry, and the advantage 

 of the latter over numbers disappears. Hence industry 

 becomes of paramount importance where insecis are scarce. 



It is, therefore, clear that there must be a strong tendency 

 for the work of cross-fertilization to be left more and more 

 to anthophilous species, to the exclusion of those insects 

 which do not entirely restrict themselves to a floral diet. 

 The visits of the latter are less valuable because of the 

 inevitable amount of revisitation. If, however, the insects 

 be very numerous in proportion to the flowers, numbers 

 may be substituted for industry without loss of efficiency, 

 and this consideration throws an interesting light on the 

 peculiarities of certain blossoms. From our present point 

 of view, entomophilous flowers may be arranged in two 

 classes, according as they facilitate the work of their 

 visitors or delay their operations. Where the number 

 of insects is small relatively to the flowers, it is to 

 the plant's advantage to expedite the work of its visitors. 

 This is, no doubt, one reason for the presence of bright 

 colours, scent, landing-stages, markings, or honey-guides, 

 as well as for colour-changes such as are seen in Lantana, 

 Hibiscus, Arnebia, and other changeable flowers, indicating 

 to visitors, as Fritz Miiller believes, those blossoms which 



